Problem
GEO-B3-M03-P009 A Point on the Chord of Contact
#9
★★☆☆☆ Level 2 of 5
B. New Original Problem. From a point \(P\), tangents to a circle \(\omega\) touch it at \(A\) and \(B\). A point \(Q\) lies on the line \(AB\). Prove that \(P\) lies on the polar of \(Q\).
Inspired by Prasolov poles and polars method
C. Hint 1. What is the polar of \(P\)?
D. Hint 2. If \(Q\) lies on the polar of \(P\), apply La Hire.
E. Full Solution.
The line \(AB\) is the polar of \(P\). By assumption \(Q\in AB\), so \(Q\) lies on the polar of \(P\).
By La Hire's theorem, \(P\) lies on the polar of \(Q\). This proves the claim.
This task is useful for recognition: every point on a chord of contact has a polar through the original external point.