thumb|upright=1.25|A triangle with [[Median (geometry)|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.]]
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". For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.
Construction
The symmedian point of a triangle with side lengths , and has homogeneous trilinear coordinates .
The mittenpunkt of a triangle is the same as the symmedian point of the excentral triangle.
The centroid of the pedal triangle of the symmedian point is the symmedian point.
Tetrahedra
For the extension to an irregular tetrahedron see symmedian.
History
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. Simon Antoine Jean L'Huilier had also noted the point in 1809.