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common chord of two intersecting circles passes through a fixed point

Source: Rioplatense Olympiad 1998 level 3 P1

September 10, 2018
geometryFixed pointarc midpointarccircles

Problem Statement

Consider an arc ABAB of a circle CC and a point PP variable in that arc ABAB. Let DD be the midpoint of the arc APAP that doeas not contain BB and let EE be the midpoint of the arc BPBP that does not contain AA. Let C1C_1 be the circle with center DD passing through AA and C2C_2 be the circle with center EE passing through B.B. Prove that the line that contains the intersection points of C1C_1 and C2C_2 passes through a fixed point.
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