Problem
GEO-B3-M04-P017 Four Simson Lines
B. New Original Problem. A quadrilateral \(ABCD\) is inscribed in a circle. Let \(l_A\) be the Simson line of point \(A\) with respect to triangle \(BCD\), and define \(l_B,l_C,l_D\) similarly. Prove that these four lines pass through one point.
C. Hint 1. Use the midpoints of the segments from a vertex to the orthocenter of the opposite triangle.
D. Hint 2. These midpoints are connected by one nine-point circle and spiral symmetry of the complete quadrilateral.
E. Full Solution.
For triangle \(BCD\), the Simson line of \(A\) passes through the midpoint of \(AH_A\), where \(H_A\) is the orthocenter of \(BCD\). The other three midpoints are defined similarly. In a cyclic quadrilateral, these four orthocenters form a system in which the corresponding midpoints lie on one nine-point circle of the complete quadrilateral.
Moreover, the directions of the four Simson lines are pairwise coordinated through the same circle: passing from one vertex to another changes the triangle and the Simson point, but preserves the common Miquel point of the orthopedal configuration. Hence all four lines pass through this point.
The last step can also be checked by angles: the intersection of \(l_A\) and \(l_B\) lies on \(l_C\) and \(l_D\), because the corresponding angles with \(AB,BC,CD,DA\) are expressed through the same arcs of the original circle.
A strong problem; the solution can be presented by angles if the Miquel point is not introduced yet.