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4

Part of 2022 Mediterranean Mathematics Olympiad

Problems(1)

PQ tangent to circumcircle of ABC

Source: Mediterranean Mathematical Olympiad 2022 P4 MMC

9/21/2022
The triangle ABCABC is inscribed in a circle γ\gamma of center OO, with AB<ACAB < AC . A point DD on the angle bisector of ∠BAC\angle BAC and a point EE on segment BCBC satisfy OEOE is parallel to ADAD and DE⊥BCDE \perp BC. Point KK lies on the extension line of EBEB such that EA=EKEA = EK. A circle pass through points A,K,DA,K,D meets the extension line of BCBC at point PP, and meets the circle of center OO at point Q≠AQ\ne A. Prove that the line PQPQ is tangent to the circle γ\gamma.
geometrytangent