OMM 1996 Problem 6

In a triangle $ABC$ with $AB < BC < AC$ , points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$ , and $CC' \perp AB$ and $CC'= AB$ , as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C'$ are collinear.

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