In the plane of any triangle ABC, let
D = midpoint of BC,
d = bisector of angle FDE,
The lines d, e, f meet in a point, labeled S in the figure. Called the Spieker center, S is the center of mass of the perimeter of triangle ABC, as proved in
Roger A. Johnson, Advanced Euclidean Geometry, Dover Publications, Mineola, NY, 1960, pp. 249-250.