Trisected perimeter point

In geometry, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that:

A´, B´, and C´ partition the perimeter of the triangle into three equal-length pieces. That is, .

The three lines AA´, BB´, and CC´ meet in a point, the trisected perimeter point.

This is point X369 in Clark Kimberling's Encyclopedia of Triangle Centers.Kimberling, C. Encyclopedia of Triangle Centers. X(369) = 1st TRISECTED PERIMETER POINT. Uniqueness and a formula for the trilinear coordinates of X369 were shown by Peter Yff late in the twentieth century. The formula involves the unique real root of a cubic equation.

See also Bisected perimeter point

References