Affine action

The Brocard point of a triangle, constructed at the intersection point of three circles.

In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845 – 1922), a French mathematician.

Definition

In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in anticlockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that

$\mathrm{\angle }PAB=\mathrm{\angle }PBC=\mathrm{\angle }PCA.$

Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle. The following applies to this angle:

$\cot \omega =\cot \alpha +\cot \beta +\cot \gamma .$

There is also a second Brocard point, Q, in triangle ABC such that line segments AQ, BQ, and CQ form equal angles with sides b, c, and a respectively. In other words, the equations $\mathrm{\angle }QCB=\mathrm{\angle }QBA=\mathrm{\angle }QAC$ apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words angle $\mathrm{\angle }PBC=\mathrm{\angle }PCA=\mathrm{\angle }PAB$ is the same as $\mathrm{\angle }QCB=\mathrm{\angle }QBA=\mathrm{\angle }QAC.$

The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken. So for example, the first Brocard point of triangle ABC is the same as the second Brocard point of triangle ACB.

The two Brocard points of a triangle ABC are isogonal conjugates of each other.

Construction

The most elegant construction of the Brocard points goes as follows. In the following example the first Brocard point is presented, but the construction for the second Brocard point is very similar.

Form a circle through points A and B, tangent to edge BC of the triangle (the center of this circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle through points B and C, tangent to edge AC, and a circle through points A and C, tangent to edge AB. These three circles have a common point, the first Brocard point of triangle ABC. See also Tangent lines to circles.

The three circles just constructed are also designated as epicycles of triangle ABC. The second Brocard point is constructed in similar fashion.

Trilinears and the Brocard midpoint

Homogeneous trilinear coordinates for the first and second Brocard points are c/b : a/c : b/a, and b/c : c/a : a/b, respectively. The Brocard points are an example of a bicentric pair of points, but they are not triangle centers because neither Brocard point is invariant under similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other. However, the unordered pair formed by both points is invariant under similarities. The midpoint of the two Brocard points, called the Brocard midpoint, has trilinears

sin(A + ω) : sin(B + ω) : sin(C + ω)

and is a triangle center. The third Brocard point, given in trilinear coordinates as a⁻³ : b⁻³ : c⁻³, or, equivalently, by

csc(A − ω) : csc(B − ω) : csc(C − ω),

is the Brocard midpoint of the anticomplementary triangle and is also the isotomic conjugate of the symmedian point.

See also

• Triangle center

Notes

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References

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  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
  
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• 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

• Third Brocard Point at MathWorld
• Bicentric Pairs of Points and Related Triangle Centers
• Bicentric Pairs of Points
• Bicentric Points at MathWorld