Problem
GEO-B2-M01-P041 A Hidden Simson Line
Point \(P\) lies on the circumcircle of an acute triangle \(ABC\). From \(P\), perpendiculars \(PX\), \(PY\), \(PZ\) are dropped to lines \(BC\), \(CA\), \(AB\), respectively. Prove that points \(X,Y,Z\) are collinear.
C. Hint 1. Find two circles passing through \(P\) and \(Y\).
D. Hint 2. Prove \(\angle XYC=\angle ZYA\), using the collinearity of \(A,Y,C\).
E. Full solution. Since \(PX\perp BC\) and \(PY\perp CA\), angles \(\angle PXC\) and \(\angle PYC\) are right angles. Therefore \(P,X,C,Y\) lie on one circle.
Similarly, \(PZ\perp AB\) and \(PY\perp CA\), so \(\angle PZA=\angle PYA=90^\circ\), and points \(P,Z,A,Y\) also lie on one circle.
From the first circle, \(\angle XYC=\angle XPC\). From the second circle, \(\angle ZYA=\angle ZPA\).
Now compare the right-hand sides. Since \(PX\perp BC\), \(\angle XPC=90^\circ-\angle BCP\). Since \(PZ\perp AB\), \(\angle ZPA=90^\circ-\angle BAP\).
Points \(A,B,C,P\) lie on one circle, so \(\angle BCP=\angle BAP\): these are inscribed angles subtending chord \(BP\). Hence \(\angle XPC=\angle ZPA\), and therefore \(\angle XYC=\angle ZYA\).
Since \(A,Y,C\) are collinear, equality \(\angle XYC=\angle ZYA\) means that rays \(YX\) and \(YZ\) lie on one line. Thus \(X,Y,Z\) are collinear.
Method comment. two circles with diameters \(PC\) and \(PA\) give equal angles at point \(Y\) Thus the solution is not a brute-force chase of all angles in the diagram, but a deliberate choice of the right circle or tangent, after which the angles can be compared through the same chord or the same line.
If the auxiliary step is skipped, the problem looks almost arbitrary: the equal angles live in different parts of the diagram. That is why the hidden configuration is identified first, then the angle replacement is made, and only at the end the required conclusion follows.
F. Difficulty justification. This is Level 9: a final-level problem with a non-obvious first step. The solution has at least three links: recognising the hidden configuration, making an oriented-angle replacement, and only then obtaining the required cyclicity, perpendicularity, or ratio.
G. Check. This is not a one-step exercise: it requires 5 key ideas. First one must recognise the hidden geometric structure, then make an angle replacement or add a circle, and only after that complete the final conclusion. For final level, it is also important that the statement gives no direct hint toward the theorem being used.