Problem
GEO-B3-M03-P018 Excircle and a Hidden Line
B. New Original Problem. The excircle of triangle \(ABC\) opposite \(A\) touches \(BC\) at \(D\), and the extensions of \(AB\) and \(AC\) at \(E\) and \(F\). Let \(T=BF\cap CE\). Prove that \(A,D,T\) are collinear.
C. Hint 1. Consider a degenerate tangential hexagon whose sides touch one circle.
D. Hint 2. Apply Brianchon to a suitable order of tangents.
E. Full Solution.
The lines \(AB\), \(AC\), \(BC\) and their extensions are tangents to the same excircle. Consider a degenerate tangential hexagon in which some neighbouring sides lie on the same lines \(AB\), \(AC\), \(BC\), and the contact points correspond to \(E,F,D\).
By Brianchon's theorem, the three diagonals of such a tangential hexagon are concurrent. Two of these diagonals give the lines meeting at \(T=BF\cap CE\), and the third is the line \(AD\). Hence \(T\in AD\).
This is a strong task on seeing Brianchon inside a triangular configuration.