Problem
GEO-B2-M04-P005 Parallelism from Two Ratios
#5
★★★☆☆ Level 3 of 5
On two rays with common endpoint \(O\), points \(A,C\) lie on the first ray and points \(B,D\) lie on the second ray. It is known that \(\frac{OA}{OC}=\frac{OB}{OD}\). Prove that \(AB \parallel CD\).
Compare triangles \(OAB\) and \(OCD\).
Triangles \(OAB\) and \(OCD\) share the angle at \(O\). The adjacent sides are proportional: \(\frac{OA}{OC}=\frac{OB}{OD}\). Therefore \(\triangle OAB \sim \triangle OCD\). From similarity, \(\angle OAB=\angle OCD\), so lines \(AB\) and \(CD\) are parallel.
A good problem for reinforcing the converse direction of homothety.