Question 11
Let \(I\) be the incenter of triangle \(ABC\), and let \(\omega\) be its circumcircle. Line \(AI\) meets \(\omega\) again at \(D\ne A\). Let \(F\) and \(E\) lie on side \(BC\) and arc \(BDC\), respectively, such that \(\angle BAF=\angle CAE\lt \tfrac12\angle BAC\). Let \(X\) be the second intersection of line \(EI\) with \(\omega\), and let \(T\) be the intersection of segment \(DX\) with line \(AF\). Prove that \(TF\cdot AD=ID\cdot AT\).