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Find the locus

Source: Spain National Olympiad 2009 (OME), Problem 6

January 23, 2010
symmetrygeometrygeometric transformationreflectiontrapezoidconicsellipse

Problem Statement

Inside a circle of center O O and radius r r, take two points A A and B B symmetrical about O O. We consider a variable point P P on the circle and draw the chord PP′‾⊥AP‾ \overline{PP'}\perp \overline{AP}. Let C C is the symmetric of B B about PP′‾ \overline{PP'} (PP‾′ \overline{PP}' is the axis of symmetry) . Find the locus of point Q \equal{} \overline{PP'}\cap\overline{AC} when we change P P in the circle.
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