Problem
GEO-B2-M01-P081 Diagonals of a Cyclic Quadrilateral
In a convex cyclic quadrilateral \(ABCD\), diagonals \(AC\) and \(BD\) meet at \(P\). It is known that \(AB=6\), \(CD=7\), and \(PD=21\). Prove that \(\triangle PAB\sim\triangle PDC\), and find \(PA\).
C. Hint 1. Do not start with products of segments; first find the similarity.
D. Hint 2. Compare \(\angle PAB\) with \(\angle PDC\), and \(\angle PBA\) with \(\angle PCD\).
E. Full solution. Since \(P,A,C\) are collinear, \(\angle PAB=\angle CAB\). Since \(P,D,B\) are collinear, \(\angle PDC=\angle CDB\).
Angles \(\angle CAB\) and \(\angle CDB\) are inscribed angles subtending the same chord \(CB\), so they are equal. Hence \(\angle PAB=\angle PDC\).
Similarly, \(\angle PBA=\angle DBA\), \(\angle PCD=\angle DCA\), and the inscribed angles \(\angle DBA\) and \(\angle DCA\) subtend chord \(DA\). Therefore \(\angle PBA=\angle PCD\).
By two angles, \(\triangle PAB\sim\triangle PDC\). From corresponding sides, \(\frac{PA}{PD}=\frac{AB}{CD}\). Hence \(PA=21\cdot\frac{6}{7}=18\).
Method comment. equal inscribed angles turn triangles \(PAB\) and \(PDC\) into similar triangles 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 6: a medium regional-style problem. A direct angle chase quickly overloads the diagram, so one must see a hidden circle or replace an angle by a tangent argument; after that, the chain is short.
G. Check. This is not a one-step exercise: it requires 3 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.