Problem
GEO-B2-M04-P002 The Centre Between Two Parallel Segments
#2
★★☆☆☆ Level 2 of 5
Lines \(AC\) and \(BD\) meet at \(O\). It is known that \(AB \parallel CD\). Prove that \(O\) is the centre of the homothety sending segment \(AB\) to segment \(CD\).
Prove that triangles \(OAB\) and \(OCD\) are similar.
Since \(AB \parallel CD\), we have \(\angle OAB=\angle OCD\) and \(\angle OBA=\angle ODC\). Hence \(\triangle OAB \sim \triangle OCD\). Thus \(\frac{OA}{OC}=\frac{OB}{OD}=\frac{AB}{CD}\). Points \(A,C,O\) and \(B,D,O\) lie on the corresponding lines, so the homothety with centre \(O\) and ratio \(\frac{OC}{OA}\) sends \(A\) to \(C\), \(B\) to \(D\), and \(AB\) to \(CD\).
The problem establishes the direct homothety criterion for two segments.