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Let P be the midpoint of the line segment AB

Source: APMO 2006, Problem 4

March 24, 2006
trigonometrygeometrycircumcirclegeometry unsolved

Problem Statement

Let A,BA,B be two distinct points on a given circle OO and let PP be the midpoint of the line segment AB. Let O1O_1 be the circle tangent to the line ABAB at PP and tangent to the circle OO. Let ll be the tangent line, different from the line ABAB, to O1O_1 passing through AA. Let CC be the intersection point, different from AA, of ll and OO. Let QQ be the midpoint of the line segment BCBC and O2O_2 be the circle tangent to the line BCBC at QQ and tangent to the line segment ACAC. Prove that the circle O2O_2 is tangent to the circle OO.
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