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Touching semicircles

Source: BdMO 2023 Secondary National P2

February 12, 2023
geometrysemicirclepower of a point

Problem Statement

Let the points A,B,CA,B,C lie on a line in this order. ABAB is the diameter of semicircle ω1\omega_1, ACAC is the diameter of semicircle ω2\omega_2. Assume both ω1\omega_1 and ω2\omega_2 lie on the same side of ACAC. DD is a point on ω2\omega_2 such that BD⊥ACBD\perp AC. A circle centered at BB with radius BDBD intersects ω1\omega_1 at EE. FF is on ACAC such that EF⊥ACEF\perp AC. Prove that BC=BFBC=BF.
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