Problem
GEO-B1-M07-P020 A Parallel for Area and Similarity
#20
★★★☆☆ Level 3 of 5
In triangle \(ABC\), point \(D\) lies on \(BC\). Through \(D\), draw a line parallel to \(AC\), meeting \(AB\) at \(E\). If \(BD:DC=1:2\), find \(S_{BDE}:S_{ABC}\).
The constructed parallel creates similar triangles.
Since \(DE\parallel AC\), \(\triangle BDE\sim\triangle BCA\). From \(BD:DC=1:2\), we get \(BD:BC=1:3\). Therefore the areas of the similar triangles are in the square of the similarity ratio: \(S_{BDE}:S_{ABC}=1:9\).
Shows that one auxiliary parallel can give both similarity and area information.