Problem
GEO-B3-M04-P013 Parallel Simson Lines
B. New Original Problem. Points \(P\) and \(Q\) lie on the circumcircle of \(ABC\). Prove that their Simson lines are parallel if and only if \(P=Q\).
C. Hint 1. The direction of the Simson line depends on the point on the circle with factor \(\frac12\).
D. Hint 2. If the directions coincide, the arcs differ by a full turn.
E. Full Solution.
When a point moves along the circumcircle by an arc of angular measure \(2\varphi\), the direction of the Simson line changes by \(\varphi\). Thus if the Simson lines of \(P\) and \(Q\) are parallel, their directions coincide modulo \(180^\circ\). Hence the arcs between \(P\) and \(Q\) have angular measure \(0^\circ\) or \(360^\circ\), so \(P=Q\).
The converse is immediate: if \(P=Q\), the Simson lines coincide, hence are parallel.
This helps avoid confusing parallelism with perpendicularity for antipodal points.