Lemoine point

In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle. In other words, it is the isogonal conjugate of the centroid of a triangle.

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

In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6). For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.