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Tangent circles and equality of segments

Source: Italian mathematical olympiad 2015, problem 5

May 12, 2015
geometryperpendicular bisector

Problem Statement

Let ABAB be a chord of a circle Γ\Gamma and let CC be a point on the segment ABAB. Let rr be a line through CC which intersects Γ\Gamma at the points D,ED,E; suppose that D,ED,E lie on different sides with respect to the perpendicular bisector of ABAB. Let ΓD\Gamma_D be the circumference which is externally tangent to Γ\Gamma at DD and touches the line ABAB at FF. Let ΓE\Gamma_E be the circumference which is externally tangent to Γ\Gamma at EE and touches the line ABAB at GG. Prove that CA=CBCA=CB if and only if CF=CGCF=CG.
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