OMM 1993 Problem 5

$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$ , the circles with diameters $OB, OC$ meet again at $X$ , and the circles with diameters $OC, OA$ meet again at $Y$ . Show that $X, Y, Z$ are collinear.