ILL 1986 Problem 57

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 $A_1,B_1, C_1$ , and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$ . Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $(A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $(A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.