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{{for|the theorem about the momentum of a force|Varignon's theorem (mechanics)}} | |||
[[Image:Varignon theorem convex.png|thumb|300px|<math> A(\Box EFGH)=\frac{1}{2}A(\Box ABCD)</math>]] | |||
'''Varignon's theorem''' is a statement in [[Euclidean geometry]] by [[Pierre Varignon]] that was first published in 1731. It deals with the construction of a particular [[parallelogram]] ('''Varignon parallelogram''') from an arbitrary [[quadrilateral|quadrangle]]. | |||
: ''The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex or reentrant, i.e. not a crossing quadrangle, then the area of the parallelogram is half as big as the area of the quadrangle''. | |||
If one introduces the concept of oriented areas for [[Polygon|n-gons]], then the area equality above holds for crossed quadrangles as well.<ref>[[Coxeter|Coxeter, H. S. M.]] and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 52–54, 1967.</ref> | |||
The Varignon parallelogram exists even for a [[Quadrilateral#More_quadrilaterals|skew quadrilateral]], and is planar whether or not the quadrilateral is planar. It can be generalized to the [[midpoint polygon]] of an arbitrary polygon. | |||
==Special cases== | |||
The Varignon parallelogram is a [[rhombus]] if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an [[equidiagonal quadrilateral]].<ref>{{citation | |||
| last = de Villiers | first = Michael | |||
| isbn = 9780557102952 | |||
| page = 58 | |||
| publisher = Dynamic Mathematics Learning | |||
| title = Some Adventures in Euclidean Geometry | |||
| url = http://books.google.com/books?id=R7uCEqwsN40C&pg=PA58 | |||
| year = 2009}}.</ref> | |||
The Varignon parallelogram is a [[rectangle]] if and only if the diagonals of the quadrilateral are [[perpendicular]], that is, if the quadrilateral is an [[orthodiagonal quadrilateral]].<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>{{rp|p.14}} | |||
==Proof== | |||
Varignon's theorem is easily proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or [[Barycentric coordinates (mathematics)|barycentric coordinates]]. The proof applies even to skew quadrilaterals in spaces of any dimension. | |||
Any three points ''E'', ''F'', ''G'' are completed to a parallelogram (lying in the plane containing ''E'', ''F'', and ''G'') by taking its fourth vertex to be ''E'' − ''F'' + ''G''. In the construction of the Varignon parallelogram this is the point (''A'' + ''B'')/2 − (''B'' + ''C'')/2 + (''C'' + ''D'')/2 = (''A'' + ''D'')/2. But this is the point ''H'' in the figure, whence ''EFGH'' forms a parallelogram. | |||
In short, the [[centroid]] of the four points ''A'', ''B'', ''C'', ''D'' is the midpoint of each of the two diagonals ''EG'' and ''FH'' of ''EFGH'', showing that the midpoints coincide. | |||
A second proof requires less mathematical aptitude. By drawing in the diagonals of the quadrilateral, we notice two triangles are created for each diagonal. And by the [[Midline Theorem]], the segment containing two midpoints of adjacent sides is both parallel and half the respective diagonal. Therefore, the sum of the diagonals is equal to the perimeter of the quadrilateral formed. Secondly, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram. | |||
{| class="wikitable" | |||
|- | |||
! convex quadrangle | |||
! reentrant quadrangle | |||
! crossed quadrangle | |||
|- | |||
|| | |||
[[Image:Varignon theorem convex.png|330px]] | |||
|| | |||
[[Image:Varignon theorem nonconvex.png|330px]] | |||
|| | |||
[[Image:Varignon theorem crossed.png|330px]] | |||
|} | |||
==See also== | |||
*[[Perpendicular bisector construction]], a different way of forming another quadrilateral from a given quadrilateral | |||
==Notes== | |||
<references/> | |||
==External links== | |||
*{{MathWorld|urlname=VarignonsTheorem|title= Varignon's theorem}} | |||
*[http://www.vias.org/comp_geometry/geom_quad_varignon.html Varignon Parallelogram in Compendium Geometry] | |||
* [http://frink.machighway.com/~dynamicm/octagoncentroids.html A generalization of Varignon's theorem to 2n-gons and to 3D] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], interactive dynamic geometry sketches. | |||
[[Category:Quadrilaterals]] | |||
[[Category:Euclidean geometry]] | |||
[[Category:Theorems in geometry]] | |||
[[Category:Articles containing proofs]] | |||
Latest revision as of 07:24, 20 August 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
Varignon's theorem is a statement in Euclidean geometry by Pierre Varignon that was first published in 1731. It deals with the construction of a particular parallelogram (Varignon parallelogram) from an arbitrary quadrangle.
- The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex or reentrant, i.e. not a crossing quadrangle, then the area of the parallelogram is half as big as the area of the quadrangle.
If one introduces the concept of oriented areas for n-gons, then the area equality above holds for crossed quadrangles as well.[1]
The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether or not the quadrilateral is planar. It can be generalized to the midpoint polygon of an arbitrary polygon.
Special cases
The Varignon parallelogram is a rhombus if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral.[2]
The Varignon parallelogram is a rectangle if and only if the diagonals of the quadrilateral are perpendicular, that is, if the quadrilateral is an orthodiagonal quadrilateral.[3]Primarily based on the most recent URA personal property value index (PPPI) flash estimates, we know that the PPPI, which represents the overall real property price development, has dipped in 2013Q4. That is the first dip the market has seen within the final 2 years.
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Proof
Varignon's theorem is easily proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates. The proof applies even to skew quadrilaterals in spaces of any dimension.
Any three points E, F, G are completed to a parallelogram (lying in the plane containing E, F, and G) by taking its fourth vertex to be E − F + G. In the construction of the Varignon parallelogram this is the point (A + B)/2 − (B + C)/2 + (C + D)/2 = (A + D)/2. But this is the point H in the figure, whence EFGH forms a parallelogram.
In short, the centroid of the four points A, B, C, D is the midpoint of each of the two diagonals EG and FH of EFGH, showing that the midpoints coincide.
A second proof requires less mathematical aptitude. By drawing in the diagonals of the quadrilateral, we notice two triangles are created for each diagonal. And by the Midline Theorem, the segment containing two midpoints of adjacent sides is both parallel and half the respective diagonal. Therefore, the sum of the diagonals is equal to the perimeter of the quadrilateral formed. Secondly, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.
| convex quadrangle | reentrant quadrangle | crossed quadrangle |
|---|---|---|
|
|
|
|
See also
- Perpendicular bisector construction, a different way of forming another quadrilateral from a given quadrilateral
Notes
- ↑ Coxeter, H. S. M. and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 52–54, 1967.
- ↑ 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.
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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.
External links
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- A generalization of Varignon's theorem to 2n-gons and to 3D at Dynamic Geometry Sketches, interactive dynamic geometry sketches.