Problem
GEO-B3-M04-P022 Simson Line of a Cyclic Quadrilateral
B. New Original Problem. A quadrilateral \(ABCD\) is cyclic, and a point \(P\) lies on the same circle. For each of the triangles \(BCD,CDA,DAB,ABC\), draw the Simson line of \(P\). Prove that the projections of \(P\) onto these four Simson lines are collinear.
C. Hint 1. Use the Simson line as an object attached to a triple of sides.
D. Hint 2. Prove the claim first for three of the four lines, then use the symmetry of the cyclic quadrilateral.
E. Full Solution.
For each triangle obtained by deleting one vertex, the Simson line of \(P\) passes through the three projections of \(P\) onto the sides of that triangle. The projection of \(P\) onto such a Simson line can be described through the circle with diameter between \(P\) and the corresponding vertex of the deleted configuration.
Consider three triangles, for example \(BCD,CDA,DAB\). The angle expressions for the projections of \(P\) onto their Simson lines depend on the same arcs of the circle \(ABCDP\). Comparing directed angles shows that three such projections are collinear. The fourth projection satisfies the same angle condition by the cyclic symmetry.
Therefore all four projections lie on one line. This line is called the Simson line of the cyclic quadrilateral with respect to \(P\).
This is a strong generalisation; it is best given after the ordinary Simson line is mastered.