Problem
GEO-B3-M03-P007 Polar of a Diagonal Point
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 that the line \(QR\) is the polar of \(P\).
C. Hint 1. For the secants \(PAB\) and \(PCD\), the corresponding tangent intersections lie on the polar of \(P\).
D. Hint 2. Use degenerate Pascal or the known self-polar diagonal triangle fact.
E. Full Solution.
Let the tangents at \(A\) and \(B\) meet at \(X\), and the tangents at \(C\) and \(D\) meet at \(Y\). By the previous fact, \(X\) and \(Y\) lie on the polar of \(P\), so the polar of \(P\) is the line \(XY\).
Apply degenerate Pascal to the circle with vertices \(A,A,C,D,B,B\). It gives that \(X,Y\) and one diagonal point are collinear; an analogous order gives the other diagonal point. Hence the line \(XY\) coincides with \(QR\). Therefore \(QR\) is the polar of \(P\).
This can be stated as a lemma: the diagonal triangle of a complete quadrangle on a circle is self-polar.