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the product MP·MQ is independent of the position of P

Source: Japan Mathematical Olympiad Finals, Problem 1

February 7, 2010
Pythagorean Theoremgeometrypower of a pointgeometry proposed

Problem Statement

Distinct points A,M,B A,M,B with AM \equal{} MB are given on circle (C0) (C_0) in this order. Let P P be a point on the arc AB AB not containing M M. Circle (C1) (C_1) is internally tangent to (C0) (C_0) at P P and tangent to AB AB at Q Q. Prove that the product MP⋅MQ MP\cdot MQ is independent of the position of P P.
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