Equal segments

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Let the triangle $ABC$ is inscribed in a circle $k(O)$. $M, N, P$ are three points of the sides $AB, BC, CA$, respectively. $k_1(O_1), k_2(O_2), k_3(O_3)$ are the circumcircles of the triangles $APM, BMN, CNP$, respectively. ${O}^{'}$ is the circumcenter of the triangle $O_1O_2O_3$. $Q$ is the intersection point of the circles $k_1, k_2, k_3.$ Prove that $O{O}^{'}={O}^{'}Q$.

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You can see two solutions of the problem:
http://www.math10.com/f/viewtopic.php?f ... 766#p49766
http://www.artofproblemsolving.com/Foru ... 8&t=499691