Problem
GEO-B3-M04-P011 Perpendicular Simson Lines
#11
★★★☆☆ Level 3 of 5
B. New Original Problem. Points \(P\) and \(Q\) are antipodal on the circumcircle of triangle \(ABC\). Prove that the Simson lines of \(P\) and \(Q\) are perpendicular.
Inspired by Prasolov Simson and pedal geometry method
C. Hint 1. Use the direction fact for the Simson line.
D. Hint 2. When passing to the antipodal point, the radius rotates by \(180^\circ\).
E. Full Solution.
The direction of the Simson line changes by half of the angular displacement of the point on the circumcircle. Points \(P\) and \(Q\) are antipodal, so the radii \(OP\) and \(OQ\) differ by a rotation of \(180^\circ\).
Therefore the directions of their Simson lines differ by \(90^\circ\). Hence the two lines are perpendicular.
This prepares the statement about the intersection point on the nine-point circle.