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maximum area involving A-excircle

Source: France 1990 P5

May 18, 2021
geometryTriangles

Problem Statement

In a triangle ABCABC, Γ\Gamma denotes the excircle corresponding to AA, A′,B′,C′A',B',C' are the points of tangency of Γ\Gamma with BC,CA,ABBC,CA,AB respectively, and S(ABC)S(ABC) denotes the region of the plane determined by segments AB′,AC′AB',AC' and the arc C′A′B′C'A'B' of Γ\Gamma.
Prove that there is a triangle ABCABC of a given perimeter pp for which the area of S(ABC)S(ABC) is maximal. For this triangle, give an approximate measure of the angle at AA.
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