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TOT 557 1997 Autumn S O2 touchpoints of incircle and excircle trisect side

Source:

September 11, 2024

geometryexcircletrisectionincircle

Problem Statement

Let

a

and

b

be two sides of a triangle. How should the third side

c

be chosen so that the points of contact of the incircle and the excircle with side

c

divide that side into three equal segments? (The excircle corresponding to the side

c

is the circle which is tangent to the side

c

and to the extensions of the sides

a

and

b

.)
(Folklore)