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4 circles, tangent in many ways, r_1+r_2+r_3=r

Source: 2002 Argentina OMA L3 p3

June 15, 2020
circlesgeometrytangent circlesradiiradius

Problem Statement

In a circumference Γ\Gamma a chord PQPQ is considered such that the segment that joins the midpoint of the smallest arc PQPQ and the midpoint of the segment PQPQ measures 11. Let Γ1,Γ2\Gamma_1, \Gamma_2 and Γ3\Gamma_3 be three tangent circumferences to the chord PQPQ that are in the same half plane than the center of Γ\Gamma with respect to the line PQPQ. Furthermore, Γ1\Gamma_1 and Γ3\Gamma_3 are internally tangent to Γ\Gamma and externally tangent toΓ2 \Gamma_2, and the centers of Γ1\Gamma_1 and Γ3\Gamma_3 are on different halfplanes with respect to the line determined by the centers of Γ\Gamma and Γ2\Gamma_2. If the sum of the radii of Γ1,Γ2\Gamma_1, \Gamma_2 and Γ3\Gamma_3 is equal to the radius of Γ\Gamma, calculate the radius of Γ2\Gamma_2.
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