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Geometry products and parallel lines

Source: Irish Mathematical Olympiad 2023 Problem 9

May 14, 2023
geometrycircumcircle

Problem Statement

The triangle ABCABC has circumcentre OO and circumcircle Γ\Gamma. Let AIAI be a diameter of Γ\Gamma. The ray AIAI extends to intersect the circumcircle ω\omega of △BOC\triangle BOC for the second time at a point PP.
Let ADAD and IQIQ be perpendicular to BCBC, with DD and QQ on BCBC. Let MM be the midpoint of BCBC.
(a) Prove that ∣AD∣⋅∣QI∣=∣CD∣⋅∣CQ∣=∣BD∣⋅∣BQ∣|AD| \cdot |QI| = |CD| \cdot |CQ| = |BD| \cdot |BQ|. (b) Prove that IMIM is parallel to PDPD.
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