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Circle and perpendicular radii.

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October 21, 2010
geometrygeometric transformationreflectiongeometry unsolved

Problem Statement

Let OO be the center of a circle. Let OU,OVOU,OV be perpendicular radii of the circle. The chord PQPQ passes through the midpoint MM of UVUV. Let WW be a point such that PM=PWPM = PW, where U,V,M,WU, V,M,W are collinear. Let RR be a point such that PR=MQPR = MQ, where RR lies on the line PWPW. Prove that MR=UVMR = UV.
Alternative version: A circle SS is given with center OO and radius rr. Let MM be a point whose distance from OO is r2\frac{r}{\sqrt{2}}. Let PMQPMQ be a chord of SS. The point NN is defined by PN→=MQ→\overrightarrow{PN} =\overrightarrow{MQ}. Let RR be the reflection of NN by the line through PP that is parallel to OMOM. Prove that MR=2rMR =\sqrt{2}r.
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