IMO 1986 LL MOR57

Origin: MOR

Problem

In a triangle ABC, the incircle touches the sides BC, CA, AB in the points A′, B′, C′, respectively; the excircle in the angle A touches the lines containing these sides in A1, B1, C1, and similarly, the excircles in the angles B and C touch these lines in A2, B2, C2 and A3, B3, C3. Prove that the triangle ABC is right-angled if and only if one of the point triples (A′, B3, C′), (A3, B′, C3), (A′, B′, C2), (A2, B2, C′), (A2, B1, C2), (A3, B3, C1), (A1, B2, C1), (A1, B1, C3) is collinear.