BrocardEllipse

Brocard ellipse

The Brocard ellipse is the conic simultaneously tangent to the sides of triangle ABC and the sides of triangle A'B'C', resulting from the intercepts of the symmedians of ABC with its circumcircle. The two triangles ABC and A'B'C' share the same symmedian point K, Brocard axis, Lemoine axis, Brocard points, Isodynamic points etc. The lines joining opposite vertices of the tangential hexagon, formed by the sides of the two triangles pass through K.
All these properties follow trivially by considering the projectivity F mapping the vertices of an equilateral A₀B₀C₀ to corresponding vertices of ABC and the center of the equilateral to the symmedian point K of the triangle.
F maps the circumcircle of the equilateral to the circumcircle of the triangle ABC.
F mas also the incircle of the equilateral to the Brocard ellipse of ABC.
The antipodal of A₀B₀C₀ is mapped by F to triangle A'B'C'. A fact from which follow the statements about the coincidence of various geometric characteristica of triangles ABC and A'B'C'.

Notice the concurrence of the four lines: tangents at opposite vertices and opposite sides (extended) at the same points of the Lemoine axis. This follows from the corresponding properties of the pre-images of these lines under projectivity F and the fact that F maps the line at infinity to the Lemoine axis of the triangle.
This is a preliminary review of a rich in content subject to be discussed in more detail at a later moment.

Brocard ellipse

See Also