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So many points

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August 29, 2010
geometrygeometry proposed

Problem Statement

In a triangle ABCABC, the incircle touches the sides BC,CA,ABBC, CA, AB in the points A′,B′,C′A',B', C', respectively; the excircle in the angle AA touches the lines containing these sides in A1,B1,C1A_1,B_1, C_1, and similarly, the excircles in the angles BB and CC touch these lines in A2,B2,C2A_2,B_2, C_2 and A3,B3,C3A_3,B_3, C_3. Prove that the triangle ABCABC is right-angled if and only if one of the point triples (A′,B3,C′),(A',B_3, C'), (A3,B′,C3),(A′,B′,C2),(A2,B2,C′),(A2,B1,C2),(A3,B3,C1), (A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1), (A1,B2,C1),(A1,B1,C3) (A_1,B_2, C_1), (A_1,B_1, C_3) is collinear.
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