Brocard Triangles

Author

Eric W. Weisstein
November 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/BrocardTriangles.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BrocardTriangles.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

First Brocard Triangle

Recognize

t=RandomTriangle[20];

pt=Cartesian[ToSidelengths[{a^2,c^2,b^2}/{a,b,c}][t]];
Select[Triangles,VertexPoints[#[t]][[1]]==pt&]

Trilinears

Triangle[BrocardTriangle[1]]

Barycentrics

(#{a,b,c}&/@Triangle[BrocardTriangle[1]][[1]])/{b c{1,1,1},a c{1,1,1},a b{1,1,1}}

Second Brocard Triangle

Recognize

pt=Cartesian[ToSidelengths[{2 SA,b^2,c^2}/{a,b,c}][t]];
Select[Triangles,VertexPoints[#[t]][[1]]==pt&]

Trilinears

Triangle[BrocardTriangle[2]]

Barycentrics

(#{a,b,c}&/@Triangle[BrocardTriangle[2]][[1]])/{a{1,1,1},b{1,1,1},c{1,1,1}}

Third Brocard Triangle

Recognize

pt=Cartesian[ToSidelengths[{b^2c^2,b^4,c^4}/{a,b,c}][t]];
Select[Triangles,VertexPoints[#[t]][[1]]==pt&]

Fourth Brocard Triangle

Recognize

pt=Cartesian[ToSidelengths[{a^2,2SA,2SA}/{a,b,c}][t]];
Select[Triangles,VertexPoints[#[t]][[1]]==pt&]

Trilinears

Triangle[DTriangle]

Barycentrics

(#{a,b,c}&/@Triangle[DTriangle][[1]]){Cos[A]{1,1,1},Cos[B]{1,1,1},Cos[C]{1,1,1}}