Problem
GEO-B3-M03-P011 Two Secants from One Point
#11
★★★☆☆ Level 3 of 5
B. New Original Problem. Through a point \(P\), two secants of a circle meet it at \(A,B\) and \(C,D\). Let \(Q=AC\cap BD\) and \(R=AD\cap BC\). Prove that \(Q\) and \(R\) lie on the polar of \(P\).
Inspired by Prasolov poles and polars method
C. Hint 1. Consider the complete quadrangle \(ABCD\).
D. Hint 2. In it \(P=AB\cap CD\), so the polar of \(P\) is the third side of the diagonal triangle.
E. Full Solution.
The points \(A,B,C,D\) lie on one circle. In the complete quadrangle \(ABCD\), the diagonal points are
\[ P=AB\cap CD,\quad Q=AC\cap BD,\quad R=AD\cap BC. \]
By self-polarity of the diagonal triangle, the polar of \(P\) is the line \(QR\). Hence both \(Q\) and \(R\) lie on the polar of \(P\).
This exercise turns the complete quadrangle lemma into a working tool.