Symmedian

The three symmedians meet at a triangle center called the Lemoine point.

Ross Honsberger has called its existence "one of the crown jewels of modern geometry".

[1] Many times in geometry, if we take three special lines through the vertices of a triangle, or cevians, then their reflections about the corresponding angle bisectors, called isogonal lines, will also have interesting properties.

For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, called the isogonal conjugate of P. The symmedians illustrate this fact.

Construct a point D by intersecting the tangents from B and C to the circumcircle.

Let the reflection of AD across the angle bisector of ∠BAC meet BC at M'.

{\displaystyle {\frac {|BM'|}{|M'C|}}={\frac {|AM'|{\frac {\sin \angle {BAM'}}{\sin \angle {ABM'}}}}{|AM'|{\frac {\sin \angle {CAM'}}{\sin \angle {ACM'}}}}}={\frac {\sin \angle {BAM'}}{\sin \angle {ACD}}}{\frac {\sin \angle {ABD}}{\sin \angle {CAM'}}}={\frac {\sin \angle {CAD}}{\sin \angle {ACD}}}{\frac {\sin \angle {ABD}}{\sin \angle {BAD}}}={\frac {|CD|}{|AD|}}{\frac {|AD|}{|BD|}}=1}

Define D' as the isogonal conjugate of D. It is easy to see that the reflection of CD about the bisector is the line through C parallel to AB.

Let ω be the circle with center D passing through B and C, and let O be the circumcenter of △ABC.

Say lines AB, AC intersect ω at P, Q, respectively.

Since we see that PQ is a diameter of ω and hence passes through D. Let M be the midpoint of BC.

Since D is the midpoint of PQ, the similarity implies that ∠BAM = ∠QAD, from which the result follows.

Let M be the midpoint of BC, and It follows that D is the Inverse of M with respect to the circumcircle.

From that, we know that the circumcircle is an Apollonian circle with foci M, D. So AS is the bisector of angle ∠DAM, and we have achieved our wanted result.

The concept of a symmedian point extends to (irregular) tetrahedra.

This is also the point that minimizes the squared distance from the faces of the tetrahedron.