Problem
GEO-B2-M04-P017 A Constructed Point and a Spiral Centre
Points \(A,C,P\) are given. Points \(B\) and \(D\) are constructed on rays \(PB\) and \(PD\) so that \(\angle APB=\angle CPD\) and \(\frac{PB}{PA}=\frac{PD}{PC}=2\). Prove that \(P\) is the centre of a spiral similarity sending \(AC\) to \(BD\).
Compare the rotation of the rays and the dilation ratio.
A rotation about \(P\) by angle \(\angle APB\) sends ray \(PA\) to ray \(PB\). By the condition, the same angle sends ray \(PC\) to ray \(PD\). Then a dilation with centre \(P\) and ratio \(2\) sends \(A\) to \(B\), because \(PB=2PA\), and sends \(C\) to \(D\), because \(PD=2PC\). Thus one rotation followed by one dilation sends \(AC\) to \(BD\), so \(P\) is the centre of the spiral similarity.
A useful constructive version of the spiral similarity criterion.