Problem
GEO-B2-M04-P015 A Point on the Parallel Image
Lines \(AC\) and \(BD\) meet at \(O\), and \(AB \parallel CD\). Point \(X\) lies on \(AB\). Line \(OX\) meets \(CD\) at \(Y\). Prove that \(\frac{OX}{OY}=\frac{OA}{OC}\).
The homothety with centre \(O\) sending \(AB\) to \(CD\) sends every point of the first segment to a point of the second.
From \(AB \parallel CD\), we get \(\triangle OAB \sim \triangle OCD\), so \(O\) is the centre of the homothety sending \(AB\) to \(CD\), with ratio \(\frac{OC}{OA}\). The image of \(X\) under this homothety lies both on line \(OX\) and on line \(CD\), so it is point \(Y\). Therefore \(\frac{OY}{OX}=\frac{OC}{OA}\), hence \(\frac{OX}{OY}=\frac{OA}{OC}\).
The problem is important: students must see that a homothety sends the whole segment, not only its endpoints.