Problem
GEO-B1-M05-P031 Perpendiculars to Two Chords
Points \(A,B,C\) lie on one circle. Line \(\ell\) is tangent to the circle at \(B\). Point \(P\) is chosen on \(\ell\). From \(P\), perpendiculars \(PX\) and \(PY\) are dropped to lines \(AB\) and \(CB\), respectively, with \(X\in AB\), \(Y\in CB\). Prove that \(XY\perp AC\).
C. Hint 1. First prove that \(P,X,B,Y\) are concyclic.
D. Hint 2. Then use the tangent-chord angle.
E. Full solution.
Since \(PX\perp AB\) and \(PY\perp CB\), we have \(\angle PXB=\angle PYB=90^\circ\). Therefore points \(P,X,B,Y\) lie on one circle with diameter \(PB\).
From this cyclic quadrilateral, \(\angle XYB=\angle XPB\), because both angles subtend chord \(XB\). Thus the angle between \(XY\) and \(CB\) equals the angle between \(XP\) and tangent \(PB\).
Since \(XP\perp AB\), the angle between \(XP\) and \(PB\) equals \(90^\circ-\angle ABP\). By the tangent-chord theorem, \(\angle ABP=\angle ACB\).
Hence the angle between \(XY\) and \(CB\) is \(90^\circ-\angle ACB\). This exactly means that \(XY\perp AC\).
A. Source analysis. Main objects: a tangent, two projections, and a cyclic quadrilateral. The obvious approach is to chase angles in the original circle, but the hidden circle passes through \(P,X,B,Y\). Number of key ideas: 3.
F. Difficulty justification. This is Level 6: a regional-style problem with one hidden cyclic quadrilateral and the tangent-chord theorem.
G. Check. This is not a one-step exercise: the perpendicularity appears only after building the second circle and transferring an angle through the tangent.