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AB=CD wanted, 3 circles of diameters XZ, XY , YZ related where X-Y-Z (collinear)

Source: 2014 Baltic Way Shortlist G4 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry

November 21, 2021
equal segmentscirclesgeometry

Problem Statement

Points XX , YY, ZZ lie on a line kk in this order. Let ω1\omega_1, ω2\omega_2, ω3\omega_3 be three circles of diameters XZXZ, XYXY , YZYZ , respectively. Line ℓ\ell passing through point YY intersects ω1\omega_1 at points AA and DD, ω2\omega_2 at BB and ω3\omega_3 at CC in such manner that points A,B,Y,X,DA, B, Y, X, D lie on ℓ\ell in this order. Prove that AB=CDAB =CD.
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