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AN x AE = AD x AF = AB x AC , 2 circles related

Source: 2014 Indonesia MO Province P2 q4 OSP

December 10, 2020
geometryProductratio

Problem Statement

Let Γ\Gamma be the circumcircle of triangle ABCABC. One circle ω\omegais tangent to Γ\Gamma at AA and tangent to BCBC at NN. Suppose that the extension of ANAN crosses Γ\Gamma again at EE. Let ADAD and AFAF be respectively the line of altitude ABCABC and diameter of Γ\Gamma, show that AN×AE=AD×AF=AB×ACAN \times AE = AD \times AF = AB \times AC
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