Problem
GEO-B2-M04-P010 One Rotation and One Ratio
Let \(\angle APB=\angle CPD\) and \(\frac{PA}{PB}=\frac{PC}{PD}\). Prove that \(P\) is the centre of a spiral similarity sending \(A\) to \(B\) and \(C\) to \(D\).
The rotation about \(P\) should send ray \(PA\) to ray \(PB\), and the dilation must have the same ratio for the second pair.
Rotate the plane about \(P\) by angle \(\angle APB\), sending ray \(PA\) to ray \(PB\). Then apply a dilation with ratio \(\frac{PB}{PA}\). Point \(A\) goes to \(B\). Since \(\angle APB=\angle CPD\), the same rotation sends ray \(PC\) to ray \(PD\). From \(\frac{PA}{PB}=\frac{PC}{PD}\), we get \(\frac{PB}{PA}=\frac{PD}{PC}\), so the same dilation sends \(C\) to \(D\). Therefore \(P\) is the centre of the spiral similarity.
Emphasise that the same composition of rotation and dilation is used.