symmedian

A symmedian of a triangle is a line through a vertex that is the isogonal conjugate of the corresponding median. Equivalently, it is obtained by reflecting the median across the angle bisector at the same vertex. For example, the A-symmedian is the line through A that is the isogonal image of the A-median.

If the A-symmedian meets BC at a point K, then the division of BC satisfies BK/CK = AB^2/AC^2. This ratio of squared adjacent sides is a hallmark property of symmedians. Another common construction is that the A-symmedian is the line through A and the intersection T of the tangents to the circumcircle at B and C; AT is the A-symmedian.

The three symmedians of triangle ABC are concurrent at a single interior point, known as the symmedian point or the Lemoine point. In barycentric coordinates relative to triangle ABC with side lengths a = BC, b = CA, c = AB, the symmedian point has coordinates a^2 : b^2 : c^2. This point serves as a notable center in several geometric and projective contexts and is the natural center associated with symmedians.