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length of MN does not depend on P and Q (KJMO 2005 p2)

Source: KJMO 2005 p2

April 30, 2019
geometrycircumcirclemidpointfixedangles

Problem Statement

For triangle ABC,PABC, P and QQ satisfy ∠BPA+∠AQC=90o\angle BPA + \angle AQC = 90^o. It is provided that the vertices of the triangle BAPBAP and ACQACQ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be NN (A≠NA \ne N, however if AA is the only intersection A=NA = N), and the midpoint of segment BCBC be MM. Show that the length of MNMN does not depend on PP and QQ.
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