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:[1] This is point X369 in Clark Kimberling's Encyclopedia of Triangle Centers.

[2] 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.

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The trisected perimeter point of a 3-4-5 right triangle. For this triangle, C´B = A´C and BA´ = CB´ , but that is not the case for triangles of other shapes.