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angle bisector wanted, 2 circumcircles related

Source: Mathematics Regional Olympiad of Mexico Center Zone 2010 P1

November 10, 2021
geometryangle bisectorcircumcircle

Problem Statement

In the acute triangle ABCABC, ∠BAC\angle BAC is less than ∠ACB\angle ACB . Let ADAD be a diameter of ω\omega, the circle circumscribed to said triangle. Let EE be the point of intersection of the ray ACAC and the tangent to ω\omega passing through BB. The perpendicular to ADAD that passes through EE intersects the circle circumscribed to the triangle BCEBCE, again, at the point FF. Show that CDCD is an angle bisector of ∠BCF\angle BCF.
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