Problem
GEO-B3-M03-P019 Full Self-Polarity
B. New Original Problem. Points \(A,B,C,D\) lie on a circle. Let \(P=AB\cap CD\), \(Q=AC\cap BD\), and \(R=AD\cap BC\). Prove all three statements: the polar of \(P\) is \(QR\), the polar of \(Q\) is \(PR\), and the polar of \(R\) is \(PQ\).
C. Hint 1. Prove one statement; the others follow by permuting the letters.
D. Hint 2. Use tangent intersections at the ends of secants.
E. Full Solution.
We prove that the polar of \(P\) is \(QR\). The pairs \(A,B\) and \(C,D\) lie on two secants through \(P\). The tangent intersections at \(A,B\) and at \(C,D\) lie on the polar of \(P\). By degenerate Pascal, the same line passes through \(Q\) and \(R\). Hence the polar of \(P\) is \(QR\).
Similarly, using the secants through \(Q\), namely \(AC\) and \(BD\), we get that the polar of \(Q\) is \(PR\). Using the secants through \(R\), namely \(AD\) and \(BC\), we get that the polar of \(R\) is \(PQ\). All three statements are proved.
After this task, the diagonal triangle can safely be used as a ready self-polar object.