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TF \cdot AD = ID \cdot AT (2016 HOMC Q11 )

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August 3, 2019
geometrycircumcircleincenter

Problem Statement

Let II be the incenter of triangle ABCABC and ω\omega be its circumcircle. Let the line AIAI intersect ω\omega at point D≠AD \ne A. Let FF and EE be points on side BCBC and arc BDCBDC respectively such that ∠BAF=∠CAE<12∠BAC\angle BAF = \angle CAE < \frac12 \angle BAC . Let XX be the second point of intersection of line EIEI with ω\omega and TT be the point of intersection of segment DXDX with line AFAF . Prove that TF⋅AD=ID⋅ATTF \cdot AD = ID \cdot AT .
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