Problem
GEO-B3-M03-P008 Pole of the Intersection of Two Polars
#8
★★☆☆☆ Level 2 of 5
B. New Original Problem. With respect to a circle \(\omega\), the polars of points \(P\) and \(Q\) meet at \(X\). Prove that the polar of \(X\) passes through \(P\) and \(Q\).
Inspired by Prasolov poles and polars method
C. Hint 1. The point \(X\) lies on the polar of \(P\).
D. Hint 2. Apply La Hire twice.
E. Full Solution.
Since \(X\) lies on the polar of \(P\), La Hire's theorem gives that \(P\) lies on the polar of \(X\). Similarly, \(X\) lies on the polar of \(Q\), so \(Q\) lies on the polar of \(X\).
The polar of \(X\) is a line. It passes through \(P\) and \(Q\), hence it is the line \(PQ\).
This is short, but it reinforces the dual transition “intersection of polars - line through poles”.