In the plane of any triangle ABC, let

D = midpoint of BC,
E = midpoint of CA,
F = midpoint of AB;

d = bisector of angle FDE,
e = bisector of angle DEF,
f = bisector of angle EFD.

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.

Triangle Centers
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