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In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal to each other.

Special cases

A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals.^[1]

A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram).

A square is a limiting case of both a kite and a rhombus.

Orthodiagonal and equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the n = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others.

Characterizations

For any orthodiagonal quadrilateral, the sum of the squares of two opposite sides equals that of the other two opposite sides: for successive sides a, b, c, and d, we have ^[2]^[3]

a^{2} + c^{2} = b^{2} + d^{2} .

This follows from the Pythagorean theorem, by which either of these two sums of two squares can be expanded to equal the sum of the four squared distances from the quadrilateral's vertices to the point where the diagonals intersect. Conversely, any quadrilateral in which a² + c² = b² + d² must be orthodiagonal.^[4] This can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers.^[5]

The diagonals of a convex quadrilateral are perpendicular if and only if the two bimedians have equal length.^[5]

According to another characterization, the diagonals of a convex quadrilateral ABCD are perpendicular if and only if

∠ P A B + ∠ P B A + ∠ P C D + ∠ P D C = π

where P is the point of intersection of the diagonals. From this equation it follows almost immediately that the diagonals of a convex quadrilateral are perpendicular if and only if the projections of the diagonal intersection onto the sides of the quadrilateral are the vertices of a cyclic quadrilateral.^[5]

A convex quadrilateral is orthodiagonal if and only if the midpoints of the sides are the vertices of a rectangle.^[5] A related characterization states that a convex quadrilateral is orthodiagonal if and only if the midpoints of the sides and the feet of the four maltitudes are eight concyclic points; the eight point circle. The center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the principal orthic quadrilateral.^[6]

If the normals to the sides of a convex quadrilateral ABCD through the diagonal intersection intersect the opposite sides in R, S, T, U, and K, L, M, N are the feet of these normals, then ABCD is orthodiagonal if and only if the eight points K, L, M, N, R, S, T and U are concyclic; the second eight point circle. A related characterization states that a convex quadrilateral is orthodiagonal if and only if RSTU is a rectangle whose sides are parallel to the diagonals of ABCD.^[5]

There are several metric characterizations regarding the four triangles formed by the diagonal intersection P and the vertices of a convex quadrilateral ABCD. Denote by m₁, m₂, m₃, m₄ the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively. If R₁, R₂, R₃, R₄ and h₁, h₂, h₃, h₄ denote the radii of the circumcircles and the altitudes respectively of these triangles, then the quadrilateral ABCD is orthodiagonal if and only if any one of the following equalities holds:^[5]

Furthermore, a quadrilateral ABCD with intersection P of the diagonals is orthodiagonal if and only if the circumcenters of the triangles ABP, BCP, CDP and DAP are the midpoints of the sides of the quadrilateral.^[5]

Comparison with a tangential quadrilateral

A few metric characterizations of tangential quadrilaterals and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table.^[5] The notations on the sides a, b, c, d, the circumradii R₁, R₂, R₃, R₄, and the altitudes h₁, h₂, h₃, h₄ are the same as above in both types of quadrilaterals.

Tangential quadrilateral	Orthodiagonal quadrilateral
$a + c = b + d$	$a^{2} + c^{2} = b^{2} + d^{2}$
$R_{1} + R_{3} = R_{2} + R_{4}$	$R_{1}^{2} + R_{3}^{2} = R_{2}^{2} + R_{4}^{2}$
$\frac{1}{h_{1}} + \frac{1}{h_{3}} = \frac{1}{h_{2}} + \frac{1}{h_{4}}$	$\frac{1}{h_{1}^{2}} + \frac{1}{h_{3}^{2}} = \frac{1}{h_{2}^{2}} + \frac{1}{h_{4}^{2}}$

Area

The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals p and q:^[7]

K = \frac{p \cdot q}{2} .

Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal.^[5] The orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals.

Other properties

Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area.^[3] For example, two rhombi both having common side a (and, as for all rhombi, both having a right angle between the diagonals), but one having a smaller acute angle than the other, have different areas (the area of the former approaching zero as the acute angle approaches zero).

If squares are erected outward on the sides of any quadrilateral (convex, concave, or crossed), then their centres (centroids) are the vertices of an orthodiagonal quadrilateral with diagonals of equal length. This is called Van Aubel's theorem.

Properties of orthodiagonal quadrilaterals that are also cyclic

Circumradius and area

For a cyclic orthodiagonal quadrilateral (one that can be inscribed in a circle), suppose the intersection of the diagonals divides one diagonal into segments of lengths p₁ and p₂ and divides the other diagonal into segments of lengths q₁ and q₂. Then^[8] (the first equality is Proposition 11 in Archimedes Book of Lemmas)

D^{2} = p_{1}^{2} + p_{2}^{2} + q_{1}^{2} + q_{2}^{2} = a^{2} + c^{2} = b^{2} + d^{2}

where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations yields that the circumradius R can be expressed as

R = \frac{1}{2} \sqrt{p_{1}^{2} + p_{2}^{2} + q_{1}^{2} + q_{2}^{2}}

or, in terms of the sides of the quadrilateral, as

R = \frac{1}{2} \sqrt{a^{2} + c^{2}} = \frac{1}{2} \sqrt{b^{2} + d^{2}} .

It also follows that

a^{2} + b^{2} + c^{2} + d^{2} = 8 R^{2} .

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

R = \sqrt{\frac{p^{2} + q^{2} + 4 x^{2}}{8}} .

A formula for the area K of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining Ptolemy's theorem and the formula for the area of an orthodiagonal quadrilateral. The result is

K = \frac{1}{2} (a c + b d) .

Other properties

In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect.^[2]

Brahmagupta's theorem states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.^[2]

If an orthodiagonal quadrilateral is also cyclic, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side.^[2]

In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.^[2]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. Republication of second edition, 1952, Barnes & Noble, pp. 136-138.
↑ ^3.0 ^3.1 Mitchell, Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ ^5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 ^5.6 ^5.7 ^5.8 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.
↑ Posamentier, Alfred S., and Charles T. Salkind, Challenging Problems in Geometry, Dover Publ., second edition, 1996:pp. 104–105, #4–23.

[1] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[Altshiller-Court-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. Republication of second edition, 1952, Barnes & Noble, pp. 136-138.

[Mitchell-3] 3.0 ^3.1 Mitchell, Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[4] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[Josefsson2-5] 5.0 ^5.1 ^5.2 ^5.3 ^5.4 ^5.5 ^5.6 ^5.7 ^5.8 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[6] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[7] Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.

[8] Posamentier, Alfred S., and Charles T. Salkind, Challenging Problems in Geometry, Dover Publ., second edition, 1996:pp. 104–105, #4–23.

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-<br><br>If you are seeking for the greatest boxed kitchen knives take into consideration Wusthof Classic Knives. Nothing at all paralyzes a knife more rapidly than the repeated attacks against the stone and glass.  Here is more information in regards to Best High-quality Japanese Chef Knives ([http://www.thebestkitchenknivesreviews.com/best-japanese-knives-chef-models-review/ thebestkitchenknivesreviews.Com]) take a look at our web-site. We asked the Examiner resident Chef, Chef Larry Edwards , if he had any such recipes.  He told us to go back into the archives on a feature  Very best Japanese Knives Set he wrote about the food getting served at the Royal Wedding and what-do-you-know, there was a perfect Old Globe side-dish just best for your Thanksgiving dinner table! Use this knife to also make your garnishes.<br><br>I have another set of Pinzon (a marker for Amazon) and the Cuisinart that I only used since they are heavier and not so sharp these knives.  I have by no means utilised a knife before and the difference in high quality is exceptional.  Based on a five-star reviews, I decided to give me, try 08 800 Sharp Presto Pro Electric Sharpener just before.  The most useful knife I've located is a 7-inch Santoku knife and 5 inches.<br><br>Chef Deagon says about chefs and tongs, We do every little thing with a pair of tongs in our hands.” The most versatile tongs are made of stainless steel and have a sturdy spring to create the helpful push that opens the tong when necessary.  A paring knife is a basic objective knife that is employed for modest intricate work like peeling vegetables, slicing smaller fruits, coring, and generating garnishes.  A paring knife generally ranges in size from three inches to five inches. Chef Deagon recommends Cuisinart®.  Great knife.<br><br>It is like a mix among a cleaver and a chef's knife and its performance is leading of the line.  The Classic Santoku Knife has a 7" blade and it is accessible in two different designs.  The knife has a pretty nice blade that is a outcome of VG-ten super steel as the core.  The knife also has a D-shaped hold, that is made with the highest quality products.  To see how the knife performs, verify out the video beneath.<br><br>July 28, 2014 place the 100 million people today of the Philippines listening to the 5th SONA of P-Noy. He catched the focus of the planet through his dance-ditty: Blurred Lines. This hub spills out the healthier things that highlight sardines as 1 of the most nutritious food in the planet. It scored a 100 Hub Score way back 2011 and gave me a satisfying one hundred % profile score.  We have all the influences amongst other top cuisines about the globe.<br><br>Rates vary, but as the time of publication, an eight-inch chef's knife retails around $60. There are three key elements to consider when picking out a chef's knife: the steel, the deal with and the weighting. Most knife handles nowadays are made from a composite material alternatively of wood and are ergonomically designed for comfort.  Look at the distinctive tasks you will be performing with your knife such as slicing, dicing and chopping.  Look for a chef's, or French, knife.
+[[File:Orthodiagonal quadrilateral.svg|thumb|240px|An orthodiagonal quadrilateral. According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides.]]
+In [[Euclidean geometry]], an '''orthodiagonal quadrilateral''' is a [[quadrilateral]] in which the [[diagonal]]s cross at [[right angle]]s. In other words, it is a four-sided figure in which the [[line segment]]s between non-adjacent [[vertex (geometry)|vertices]] are [[orthogonal]] to each other.
+==Special cases==
+A [[kite (geometry)|kite]] is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a [[circle]] tangent to all four of their sides; that is, the kites are the [[tangential quadrilateral|tangential]] orthodiagonal quadrilaterals.<ref>{{citation
+ | last = Josefsson | first = Martin
+ | journal = Forum Geometricorum
+ | pages = 119–130
+ | title = Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral
+ | url = http://forumgeom.fau.edu/FG2010volume10/FG201013.pdf
+ | volume = 10
+ | year = 2010}}.</ref>
+A [[rhombus]] is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a [[parallelogram]]).
+A [[square (geometry)|square]] is a limiting case of both a kite and a rhombus.
+Orthodiagonal and [[Equidiagonal quadrilateral|equidiagonal]] quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the ''n''&nbsp;=&nbsp;4 case of the [[biggest little polygon]] problem. The square is one such quadrilateral, but there are infinitely many others.
+==Characterizations==
+For any orthodiagonal quadrilateral, the sum of the squares of two opposite sides equals that of the other two opposite sides: for successive sides ''a'', ''b'', ''c'', and ''d'', we have <ref name=Altshiller-Court>{{citation|last=Altshiller-Court|first=N.|title=College Geometry|publisher=Dover Publications|year=2007}}. Republication of second edition, 1952, Barnes & Noble, pp. 136-138.</ref><ref name=Mitchell>Mitchell, {{citation|last=Douglas|first=W.|title=The area of a quadrilateral|journal=Mathematical Gazette|volume=93|year=2009|pages=306–309}}.</ref>
+:<math>\displaystyle a^2+c^2=b^2+d^2. </math>
+This follows from the [[Pythagorean theorem]], by which either of these two sums of two squares can be expanded to equal the sum of the four squared distances from the quadrilateral's vertices to the point where the diagonals intersect. Conversely, any quadrilateral in which ''a''<sup>2</sup> + ''c''<sup>2</sup> = ''b''<sup>2</sup> + ''d''<sup>2</sup> must be orthodiagonal.<ref>{{citation|url=http://forumgeom.fau.edu/FG2009volume9/FG200919.pdf|title=Class preserving dissections of convex quadrilaterals|journal=Forum Geometricorum|volume=9|year=2009|pages=195–211|first1=Dan|last1=Ismailescu|first2=Adam|last2=Vojdany}}.</ref> This can be proved in a number of ways, including using the [[law of cosines]], [[Euclidean vector|vectors]], an [[Proof by contradiction|indirect proof]], and [[complex number]]s.<ref name=Josefsson2>{{citation
+ | last = Josefsson | first = Martin
+ | journal = Forum Geometricorum
+ | pages = 13–25
+ | title = Characterizations of Orthodiagonal Quadrilaterals
+ | url = http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf
+  | volume = 12
+ | year = 2012}}.</ref>
+The diagonals of a convex quadrilateral are perpendicular [[if and only if]] the two [[Quadrilateral#Special line segments|bimedians]] have equal length.<ref name=Josefsson2/>
+According to another characterization, the diagonals of a convex quadrilateral ''ABCD'' are perpendicular if and only if
+:<math>\angle PAB + \angle PBA + \angle PCD + \angle PDC = \pi</math>
+where ''P'' is the point of intersection of the diagonals. From this equation it follows almost immediately that the diagonals of a convex quadrilateral are perpendicular if and only if the [[Projection (mathematics)|projections]] of the diagonal intersection onto the sides of the quadrilateral are the vertices of a [[cyclic quadrilateral]].<ref name=Josefsson2/>
+A convex quadrilateral is orthodiagonal if and only if the [[midpoint]]s of the sides are the vertices of a [[rectangle]].<ref name=Josefsson2/> A related characterization states that a convex quadrilateral is orthodiagonal if and only if the midpoints of the sides and the feet of the four [[Quadrilateral#Special line segments|maltitudes]] are eight [[concyclic points]]; the ''eight point circle''. The center of this circle is the [[Quadrilateral#Special line segments|centroid]] of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the ''principal orthic quadrilateral''.<ref>{{citation
+ | last1 = Mammana | first1 = Maria Flavia | last2 = Micale | first2 = Biagio | last3 = Pennisi | first3 = Mario
+ | journal = Forum Geometricorum
+ | pages = 109–119
+ | title = The Droz-Farny Circles of a Convex Quadrilateral
+ | url = http://forumgeom.fau.edu/FG2011volume11/FG201111.pdf
+  | volume = 11
+ | year = 2011}}.</ref>
+If the [[Normal (geometry)|normals]] to the sides of a convex quadrilateral ''ABCD'' through the diagonal intersection intersect the opposite sides in ''R'', ''S'', ''T'', ''U'', and ''K'', ''L'', ''M'', ''N'' are the feet of these normals, then ''ABCD'' is orthodiagonal if and only if the eight points ''K'', ''L'', ''M'', ''N'', ''R'', ''S'', ''T'' and ''U'' are concyclic; the ''second eight point circle''. A related characterization states that a convex quadrilateral is orthodiagonal if and only if ''RSTU'' is a rectangle whose sides are [[Parallel (geometry)|parallel]] to the diagonals of ''ABCD''.<ref name=Josefsson2/>
+There are several metric characterizations regarding the four [[triangle]]s formed by the diagonal intersection ''P'' and the vertices of a convex quadrilateral ''ABCD''. Denote by ''m''<sub>1</sub>, ''m''<sub>2</sub>, ''m''<sub>3</sub>, ''m''<sub>4</sub> the [[Median (geometry)|medians]] in triangles ''ABP'', ''BCP'', ''CDP'', ''DAP'' from ''P'' to the sides ''AB'', ''BC'', ''CD'', ''DA'' respectively. If ''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, ''R''<sub>4</sub> and ''h''<sub>1</sub>, ''h''<sub>2</sub>, ''h''<sub>3</sub>, ''h''<sub>4</sub> denote the [[radius|radii]] of the [[circumcircle]]s and the [[Altitude (triangle)|altitude]]s respectively of these triangles, then the quadrilateral ''ABCD'' is orthodiagonal if and only if any one of the following equalities holds:<ref name=Josefsson2/>
+* <math>m_1^2+m_3^2=m_2^2+m_4^2</math>
+* <math>R_1^2+R_3^2=R_2^2+R_4^2</math>
+* <math>\frac{1}{h_1^2}+\frac{1}{h_3^2}=\frac{1}{h_2^2}+\frac{1}{h_4^2}</math>
+Furthermore, a quadrilateral ''ABCD''  with intersection ''P'' of the diagonals is orthodiagonal if and only if the circumcenters of the triangles ''ABP'', ''BCP'', ''CDP'' and ''DAP'' are the midpoints of the sides of the quadrilateral.<ref name=Josefsson2/>
+===Comparison with a tangential quadrilateral===
+A few metric characterizations of [[tangential quadrilateral]]s and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table.<ref name=Josefsson2/> The notations on the sides ''a'', ''b'', ''c'', ''d'', the circumradii ''R''<sub>1</sub>, ''R''<sub>2</sub>, ''R''<sub>3</sub>, ''R''<sub>4</sub>, and the altitudes ''h''<sub>1</sub>, ''h''<sub>2</sub>, ''h''<sub>3</sub>, ''h''<sub>4</sub> are the same as above in both types of quadrilaterals.
+{| class=wikitable
+|-
+! Tangential quadrilateral
+! Orthodiagonal quadrilateral
+|-
+| align=center|<math>a+c=b+d</math>
+| align=center|<math>a^2+c^2=b^2+d^2</math>
+|-
+| align=center|<math>R_1+R_3=R_2+R_4</math>
+| align=center|<math>R_1^2+R_3^2=R_2^2+R_4^2</math>
+|-
+| align=center|<math>\frac{1}{h_1}+\frac{1}{h_3}=\frac{1}{h_2}+\frac{1}{h_4}</math>
+| align=center|<math>\frac{1}{h_1^2}+\frac{1}{h_3^2}=\frac{1}{h_2^2}+\frac{1}{h_4^2}</math>
+|}
+==Area==
+The area ''K'' of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals ''p'' and ''q'':<ref>Harries, J. "Area of a quadrilateral," ''Mathematical Gazette'' 86, July 2002, 310–311.</ref>
+:<math>K=\frac{p\cdot q}{2}.</math>
+Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal.<ref name=Josefsson2/> The orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals.
+==Other properties==
+*Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area.<ref name=Mitchell/>  For example, two rhombi both having common side ''a'' (and, as for all rhombi, both having a right angle between the diagonals), but one having a smaller [[acute angle]] than the other, have different areas (the area of the former approaching zero as the acute angle approaches zero).
+*If [[Square (geometry)|square]]s are erected outward on the sides of any [[quadrilateral]] (convex, concave, or crossed), then their [[Centre (geometry)|centre]]s ([[centroid]]s) are the vertices of an orthodiagonal quadrilateral with diagonals of equal length. This is called [[Van Aubel's theorem]].
+==Properties of orthodiagonal quadrilaterals that are also cyclic==
+===Circumradius and area===
+For a [[Cyclic quadrilateral|cyclic]] orthodiagonal quadrilateral (one that can be [[inscribed]] in a [[circle]]), suppose the intersection of the diagonals divides one diagonal into segments of lengths ''p''<sub>1</sub> and ''p''<sub>2</sub> and divides the other diagonal into segments of lengths ''q''<sub>1</sub> and ''q''<sub>2</sub>. Then<ref>Posamentier, Alfred S., and Charles T. Salkind, ''Challenging Problems in Geometry'', Dover Publ., second edition, 1996:pp. 104–105, #4–23.</ref> (the first equality is Proposition 11 in [[Archimedes]] [[Book of Lemmas]])
+:<math> D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2 </math>
+where ''D'' is the [[diameter]] of the [[circumcircle]].  This holds because the diagonals are perpendicular [[Circle#Chord|chords of a circle]]. These equations yields that the [[circumradius]] ''R'' can be expressed as
+:<math> R=\tfrac{1}{2}\sqrt{p_1^2+p_2^2+q_1^2+q_2^2} </math>
+or, in terms of the sides of the quadrilateral, as
+:<math> R=\tfrac{1}{2}\sqrt{a^2+c^2}=\tfrac{1}{2}\sqrt{b^2+d^2}. </math>
+It also follows that
+:<math> a^2+b^2+c^2+d^2=8R^2. </math>
+Thus, according to [[Quadrilateral#Diagonals|Euler's quadrilateral theorem]], the circumradius can be expressed in terms of the diagonals ''p'' and ''q'', and the distance ''x'' between the midpoints of the diagonals as
+:<math> R=\sqrt{\frac{p^2+q^2+4x^2}{8}}. </math>
+A formula for the [[area]] ''K'' of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining [[Cyclic quadrilateral#Diagonals|Ptolemy's theorem]] and the formula for the [[Orthodiagonal quadrilateral#Area|area of an orthodiagonal quadrilateral]]. The result is
+:<math> K=\tfrac{1}{2}(ac+bd). </math>
+===Other properties===
+*In a cyclic orthodiagonal quadrilateral, the [[Cyclic quadrilateral#Other properties|anticenter]] coincides with the point where the diagonals intersect.<ref name=Altshiller-Court/>
+*[[Brahmagupta's theorem]] states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.<ref name=Altshiller-Court/>
+*If an orthodiagonal quadrilateral is also cyclic, the distance from the [[circumcenter]] (the center of the circumscribed circle) to any side equals half the length of the opposite side.<ref name=Altshiller-Court/>
+*In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.<ref name=Altshiller-Court/>
+==References==
+{{reflist}}
+[[Category:Quadrilaterals]]

Well-structured transition system: Difference between revisions

Latest revision as of 11:55, 13 March 2013

Contents

Special cases

Characterizations

Comparison with a tangential quadrilateral

Area

Other properties

Properties of orthodiagonal quadrilaterals that are also cyclic

Circumradius and area

Other properties

References

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Well-structured transition system: Difference between revisions

Latest revision as of 11:55, 13 March 2013

Special cases

Characterizations

Comparison with a tangential quadrilateral

Area

Other properties

Properties of orthodiagonal quadrilaterals that are also cyclic

Circumradius and area

Other properties

References

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