Problem
GEO-B3-M03-P006 Tangents at the Ends of a Secant
#6
★★☆☆☆ Level 2 of 5
B. New Original Problem. A secant through a point \(P\) meets a circle \(\omega\) at \(A\) and \(B\). The tangents to \(\omega\) at \(A\) and \(B\) meet at \(T\). Prove that \(T\) lies on the polar of \(P\).
Inspired by Prasolov poles and polars method
C. Hint 1. What is the polar of \(T\)?
D. Hint 2. Since \(P\in AB\), apply La Hire.
E. Full Solution.
Since \(TA\) and \(TB\) are tangents, the polar of \(T\) is the line \(AB\). The point \(P\) lies on \(AB\), so \(P\) lies on the polar of \(T\).
By La Hire's theorem, \(T\) lies on the polar of \(P\), as required.
This is the key fact for all problems where several secants pass through one point.