Brokar's Triangle

Brokar's triangle (black) for the triangle

\triangle ABC

{\ displaystyle \ triangle ABC}

\ triangle ABC

,

B_{1}

{\ displaystyle B_ {1}}

B_1

and

B_{2}

{\ displaystyle B_ {2}}

B_ {2}

- two points of Brokar.

Brokar Triangle - a triangle formed by the points of intersection of lines drawn from two different vertices of a given triangle through different Brokar points : for $\triangle ABC$ ${\ displaystyle \ triangle ABC}$ $\ triangle ABC$ and its Brokar points $B_{1}$ ${\ displaystyle B_ {1}}$ $B_1$ and $B_{2}$ ${\ displaystyle B_ {2}}$ $B_ {2}$ the vertices of one of Brokar's triangles will be at the intersections $AB_{1}\cap BB_{2}$ ${\ displaystyle AB_ {1} \ cap BB_ {2}}$ ${\ displaystyle AB_ {1} \ cap BB_ {2}}$ , $AB_{1}\cap CB_{2}$ ${\ displaystyle AB_ {1} \ cap CB_ {2}}$ ${\ displaystyle AB_ {1} \ cap CB_ {2}}$ and $AB_{2}\cap BB_{1}$ ${\ displaystyle AB_ {2} \ cap BB_ {1}}$ ${\ displaystyle AB_ {2} \ cap BB_ {1}}$ ^[1] . Brokar's triangle is inscribed in the circle of Brokar ^[2] .

Content

History

Named after the French meteorologist and geometer Henri Brocard ^[3] .

Another way to build Brokar's triangle

A line passing through A is parallel to B'C ' , a line passing through B is parallel to C'A' , and a line passing through C is parallel to A'B ' intersect at the Steiner point .

Brokar's triangle can be constructed as follows.

Let triangle ABC be given. Let O be its center of the circumscribed circle and K be the intersection point of the simedians of triangle ABC . The circle constructed on OK , as on the diameter, represents the Brokar circle of triangle ABC . a line passing through O perpendicular to the line BC intersects the Brocard circle at another point A ' . A line passing through O perpendicular to the line CA intersects the Brocard circle at another point B ' . A line passing through O perpendicular to straight AB intersects the Brocard circle at another point C ' . Triangle A'B'C ' is Brokars triangle for triangle ABC .

Notes

↑ Gentry, FC (1941), "Analytic geometry of the triangle", National Mathematics Magazine T. 16: 127–140
↑ Weisstein, Eric W. First Brocard Triangle on the Wolfram MathWorld website.
↑ Brocard biography

[1] Gentry, FC (1941), "Analytic geometry of the triangle", National Mathematics Magazine T. 16: 127–140

[2] Weisstein, Eric W. First Brocard Triangle on the Wolfram MathWorld website.

[3] Brocard biography

Brokar's Triangle

Content

History

Another way to build Brokar's triangle

See also

Notes

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