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prove midpoints

Source: Pan-American Girls’ Mathematical Olympiad 2023 P4

August 9, 2023
geometry

Problem Statement

In an acute-angled triangle ABCABC, let DD be a point on the segment BCBC. Let RR and SS be the feet of the perpendiculars from DD to ACAC and ABAB, respectively. The line DRDR intersects the circumcircle of BDSBDS at XX, with X≠DX \neq D. Similarly, the line DSDS intersects the circumcircle of CDRCDR at YY, with Y≠DY \neq D. Prove that if XYXY is parallel to RSRS, then DD is the midpoint of BCBC.
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