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|CE|x|DG| = |EF| x |CG| for inscribed right isosceles triangle

Source: 2021 Irish Mathematical Olympiad P2

May 30, 2021
geometryratioright triangleisosceles

Problem Statement

An isosceles triangle ABCABC is inscribed in a circle with ∠ACB=90o\angle ACB = 90^o and EFEF is a chord of the circle such that neither E nor FF coincide with CC. Lines CECE and CFCF meet ABAB at DD and GG respectively. Prove that ∣CE∣⋅∣DG∣=∣EF∣⋅∣CG∣|CE|\cdot |DG| = |EF| \cdot |CG|.
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