Lemoine point

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

[2] For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.

[2] An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the hesse normal forms of the corresponding lines.

It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides.

Line AA' is a symmedian, as can be seen by drawing the circle with center A' through B and C.[citation needed] The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it in 1847.

A triangle with medians (black), angle bisectors (dotted) and symmedians (red). The symmedians intersect in the symmedian point L, the angle bisectors in the incenter I and the medians in the centroid G.