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SAMO Problem 3: Product of lengths in triangle with $45^\circ$ angle

Source: South African Mathematics Olympiad 2019, Problem 3

July 25, 2019
geometry

Problem Statement

Let AA, BB, CC be points on a circle whose centre is OO and whose radius is 11, such that ∠BAC=45∘\angle BAC = 45^\circ. Lines ACAC and BOBO (possibly extended) intersect at DD, and lines ABAB and COCO (possibly extended) intersect at EE. Prove that BD⋅CE=2BD \cdot CE = 2.
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