Problem
GEO-B3-M04-P020 Rotation of the Simson Line
B. New Original Problem. A point \(P\) moves along an arc of the circumcircle of \(ABC\) from \(P_1\) to \(P_2\), and the central angle \(\angle P_1OP_2=2\varphi\). Prove that the angle between the Simson lines of \(P_1\) and \(P_2\) is \(\varphi\).
C. Hint 1. Express the direction of the Simson line through an inscribed angle with vertex on the circle.
D. Hint 2. An inscribed angle equals half of the corresponding central angle.
E. Full Solution.
The direction of the Simson line of \(P\) can be expressed through the angle between this line and a fixed side, say \(AB\). In the proof of the Simson theorem, this angle equals an inscribed angle depending on an arc with endpoint \(P\).
When \(P\) moves from \(P_1\) to \(P_2\), the corresponding central angle changes by \(2\varphi\), while the inscribed angle changes by \(\varphi\). Hence the direction of the Simson line changes by \(\varphi\). Therefore the angle between the two Simson lines is \(\varphi\).
This is the main dynamic fact for families of Simson lines.