Grebe point | Lemoine point

The point of concurrence K of the symmedians, sometimes also called the Lemoine point (in England and France) or the Grebe point (in Germany). Equivalently, the symmedian point is the isogonal conjugate of the triangle centroid G. In other words, let G be the triangle centroid of a triangle Δ A B C, A M_A, B M_B, and C M_C the medians of Δ A B C, A L_A, B L_B, and C L_C the angle bisectors of angles A, B, C, and A K_A, B K_B, and C K_C the reflections of A M_A, B M_B, and C M_C about A L_A, B L_B, and C L_C. Then K is the point of concurrence of the lines A K_A, B K_B, and C K_C. According to Honsberger, the symmedian point is "one of the crown jewels of modern geometry." The symmedian point is Kimberling center X_6.

angle bisector | Brocard angle | Brocard axis | Brocard diameter | cosymmedian triangles | first Lemoine circle | isogonal conjugate | Lemoine axis | line at infinity | mittenpunkt | pedal triangle | Schoute center | Steiner points | symmedian | tangential triangle | triangle centroid