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AB = 2CD wanted, circle tangent to chord and another circle

Source: 2021 Saudi Arabia Training Lists p10 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022
geometryequal segmentstangent circles

Problem Statement

Let ABAB be a chord of the circle (O)(O). Denote M as the midpoint of the minor arc ABAB. A circle (O′)(O') tangent to segment ABAB and internally tangent to (O)(O). A line passes through MM, perpendicular to O′AO'A, O′BO'B and cuts ABAB respectively at C,DC, D. Prove that AB=2CDAB = 2CD.
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