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sequence of incircle's triangles

Source: Israel Grosman Memorial Mathematical Olympiad 1999 p3

February 15, 2020
incirclegeometryEquilateral

Problem Statement

For every triangle ABCABC, denote by D(ABC)D(ABC) the triangle whose vertices are the tangency points of the incircle of △ABC\vartriangle ABC with the sides. Assume that △ABC\vartriangle ABC is not equilateral. (a) Prove that D(ABC)D(ABC) is also not equilateral. (b) Find in the sequence T1=△ABC,Tk+1=D(Tk)T_1 = \vartriangle ABC, T_{k+1} = D(T_k) for k∈Nk \in N a triangle whose largest angle α\alpha satisfies 0<α−60o<0.0001o0 < \alpha -60^o < 0.0001^o
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