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PQ = r and 6 more conditions

Source: 2023 ISL G2

July 17, 2024
geometry

Problem Statement

Let ABCABC be a triangle with AC>BC,AC > BC, let ω\omega be the circumcircle of △ABC,\triangle ABC, and let rr be its radius. Point PP is chosen on AC‾\overline{AC} such taht BC=CP,BC=CP, and point SS is the foot of the perpendicular from PP to AB‾\overline{AB}. Ray BPBP mets ω\omega again at DD. Point QQ is chosen on line SPSP such that PQ=rPQ = r and S,P,QS,P,Q lie on a line in that order. Finally, let EE be a point satisfying AE‾⊥CQ‾\overline{AE} \perp \overline{CQ} and BE‾⊥DQ‾\overline{BE} \perp \overline{DQ}. Prove that EE lies on ω\omega.
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