Problem
GEO-B3-M04-P016 Simson Line and a Parallel to an Altitude
B. New Original Problem. Let \(P\) lie on the circumcircle of \(ABC\), and let \(A_1,B_1,C_1\) be its projections onto \(BC,CA,AB\). Prove that if \(PA\parallel BC\), then the Simson line \(A_1B_1C_1\) is parallel to the altitude from \(A\).
C. Hint 1. Use the direction of the Simson line through its angle with \(AB\) or \(AC\).
D. Hint 2. The condition \(PA\parallel BC\) converts an angle at \(P\) into a triangle angle.
E. Full Solution.
By the Simson theorem, \(A_1,B_1,C_1\) are collinear. Consider the angle between the Simson line and \(AB\). Through the cyclic quadrilateral \(P,B_1,A,C_1\), it equals the angle between \(PB_1\) and \(PA\), that is, the angle between a perpendicular to \(AC\) and a line parallel to \(BC\).
This angle equals the angle between the altitude from \(A\) and \(AB\). Therefore the Simson line has the same direction as the altitude from \(A\), so it is parallel to it.
A good direction problem without heavy rotation theory of the Simson line.