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Difference between revisions of "Brocard point"

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The '''Brocard point''' of a [[triangle]] is the point <math>P</math> in triangle <math>\triangle ABC</math> such that <math>\angle PAB=\angle PCA=\angle PBC</math>. It is also the unique point <math>P</math> inside <math>\triangle ABC</math> such that the sum of the distances from <math>P</math> to <math>A, B,</math> and <math>C</math> is a minimum. These points are named after French mathematician [[Henri Brocard]].
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There are two '''Brocard points''' within any [[triangle]]. The '''first Brocard point''' is the point <math>P</math> in triangle <math>\triangle ABC</math> labeled in ''counterclockwise'' order such that <math>\angle PAB=\angle PCA=\angle PBC</math>, with the unique angle denoted as <math>\omega</math>, the '''Brocard angle'''. The '''second Brocard point''' is a point <math>Q</math> inside <math>\triangle ABC</math> such that <math>\angle QAC=\angle QCA=\angle QBA</math>. Moreover, these two triples of angles are equal. In general, we have <cmath>\angle PAB=\angle PCA=\angle PBC=\angle QAC=\angle QCA=\angle QBA=\omega</cmath> The two Brocard points of a triangle only coincide when the triangle is equilateral. These points are named after French mathematician [[Henri Brocard]].
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The Brocard angle <math>\omega</math> is given by the identities:
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<cmath>\begin{align*}\cot(\omega)&=\cot(A)+\cot(B)+\cot(C)\\&=\frac{a^2+b^2+c^2}{4\triangle}\end{align*}</cmath>
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where <math>\triangle</math> is the area of triangle <math>\triangle ABC</math> with <math>\angle A, \angle B, \angle C</math> opposite side <math>a, b, c</math>, respectively.  
  
 
== Problems ==
 
== Problems ==

Latest revision as of 00:41, 7 September 2025

There are two Brocard points within any triangle. The first Brocard point is the point $P$ in triangle $\triangle ABC$ labeled in counterclockwise order such that $\angle PAB=\angle PCA=\angle PBC$, with the unique angle denoted as $\omega$, the Brocard angle. The second Brocard point is a point $Q$ inside $\triangle ABC$ such that $\angle QAC=\angle QCA=\angle QBA$. Moreover, these two triples of angles are equal. In general, we have \[\angle PAB=\angle PCA=\angle PBC=\angle QAC=\angle QCA=\angle QBA=\omega\] The two Brocard points of a triangle only coincide when the triangle is equilateral. These points are named after French mathematician Henri Brocard.

The Brocard angle $\omega$ is given by the identities: \begin{align*}\cot(\omega)&=\cot(A)+\cot(B)+\cot(C)\\&=\frac{a^2+b^2+c^2}{4\triangle}\end{align*} where $\triangle$ is the area of triangle $\triangle ABC$ with $\angle A, \angle B, \angle C$ opposite side $a, b, c$, respectively.

Problems

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