Problem
GEO-B3-M04-P009 A Chord Perpendicular to a Side
B. New Original Problem. On the circumcircle of \(ABC\), points \(P\) and \(Q\) are such that the chord \(PQ\perp BC\). Prove that the Simson line of \(P\) is parallel to \(AQ\).
C. Hint 1. Use the direction of the Simson line through inscribed angles.
D. Hint 2. The condition \(PQ\perp BC\) translates the direction through an arc ending at \(Q\).
E. Full Solution.
Let \(A_1,B_1,C_1\) be the projections of \(P\) onto \(BC,CA,AB\). The direction of the Simson line can be expressed, for instance, through the angle between \(B_1C_1\) and \(AB\). From the cyclicity of \(P,B_1,A,C_1\), this angle equals the corresponding angle between \(PA\) and \(PC\).
Since \(PQ\perp BC\), the inscribed angles defining the direction of \(AQ\) coincide with the angles defining the direction of \(B_1C_1\). Therefore \(B_1C_1\parallel AQ\). Hence the whole Simson line of \(P\) is parallel to \(AQ\).
A good first step toward direction problems for the Simson line.