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A point of tangency, a locus and orthogonal circles

Source: Iberoamerican Olympiad 1990, Problem 4

May 21, 2007
conicsparabolageometryrectanglepower of a pointradical axisgeometry proposed

Problem Statement

Let Γ1\Gamma_{1} be a circle. ABAB is a diameter, ℓ\ell is the tangent at BB, and MM is a point on Γ1\Gamma_{1} other than AA. Γ2\Gamma_{2} is a circle tangent to ℓ\ell, and also to Γ1\Gamma_{1} at MM. a) Determine the point of tangency PP of ℓ\ell and Γ2\Gamma_{2} and find the locus of the center of Γ2\Gamma_{2} as MM varies. b) Show that there exists a circle that is always orthogonal to Γ2\Gamma_{2}, regardless of the position of MM.
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