Problem
GEO-B3-M03-P013 Pole of a Side in a Triangle with Incircle
B. New Original Problem. The incircle of triangle \(ABC\) touches \(AB\) and \(AC\) at \(F\) and \(E\). Prove that the line \(EF\) is the polar of \(A\) with respect to the incircle. Then prove that if a point \(X\) lies on \(EF\), the polar of \(X\) passes through \(A\).
C. Hint 1. The sides \(AB\) and \(AC\) are tangents from \(A\).
D. Hint 2. After that, apply La Hire to the points \(A\) and \(X\).
E. Full Solution.
Since \(AB\) and \(AC\) touch the incircle at \(F\) and \(E\), the line \(EF\), joining the contact points of the two tangents from \(A\), is the polar of \(A\).
If \(X\in EF\), then \(X\) lies on the polar of \(A\). By La Hire's theorem, \(A\) lies on the polar of \(X\). Hence the polar of \(X\) passes through \(A\).
A convenient task for moving from an abstract circle to triangles with contact points.