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Circle centered on A-excenter passing A

Source: Korea National Olympiad 2019 P2

November 16, 2019
geometrycircumcircle

Problem Statement

Triangle ABCABC is an scalene triangle. Let II the incenter, Ω\Omega the circumcircle, EE the AA-excenter of triangle ABCABC. Let Γ\Gamma the circle centered at EE and passes AA. Γ\Gamma and Ω\Omega intersect at point D(≠A)D(\neq A), and the perpendicular line of BCBC which passes AA meets Γ\Gamma at point K(≠A)K(\neq A). LL is the perpendicular foot from II to ACAC. Now if AEAE and DKDK intersects at FF, prove that BE⋅CI=2⋅CF⋅CLBE\cdot CI=2\cdot CF\cdot CL.
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