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Line passes through midpoint

Source: Centroamerican 2012, Problem 2

June 19, 2012
geometrycircumcircletrapezoidperpendicular bisectorgeometry unsolved

Problem Statement

Let γ\gamma be the circumcircle of the acute triangle ABCABC. Let PP be the midpoint of the minor arc BCBC. The parallel to ABAB through PP cuts BC,ACBC, AC and γ\gamma at points R,SR,S and TT, respectively. Let K≡AP∩BTK \equiv AP \cap BT and L≡BS∩ARL \equiv BS \cap AR. Show that KLKL passes through the midpoint of ABAB if and only if CS=PRCS = PR.
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