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Common point on line XO

Source: Junior Olympiad of Malaysia 2014 P5

July 20, 2015
geometry

Problem Statement

Given △ABC\triangle ABC with circumcircle Γ\Gamma and circumcentre OO, let XX be a point on Γ\Gamma. Let XC1XC_1, XB1XB_1 to be feet of perpendiculars from XX to lines ABAB and ACAC. Define ωC\omega_C as the circle with centre the midpoint of ABAB and passing through C1C_1 . Define ωB\omega_B similarly.
Prove that ωB\omega_B and ωC\omega_C has a common point on XOXO.
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