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2017 G9: Concurrency in Cyclic Quadrilateral

Source:

January 29, 2017
2017geometrycyclic quadrilateral

Problem Statement

Let △ABC\triangle ABC be an acute triangle with circumcenter OO, and let Q≠AQ\neq A denote the point on ⊙(ABC)\odot (ABC) for which AQ⊥BCAQ\perp BC. The circumcircle of △BOC\triangle BOC intersects lines ACAC and ABAB for the second time at DD and EE respectively. Suppose that AQAQ, BCBC, and DEDE are concurrent. If OD=3OD=3 and OE=7OE=7, compute AQAQ.
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