Problem
GEO-B2-M02-P013 A Product on the Sides of a Triangle
#13
★★★★☆ Level 4 of 5
In triangle \(ABC\), points \(D\in AB\) and \(E\in AC\) are such that \(B,C,D,E\) lie on one circle. Prove that \(\frac{AD}{AE}=\frac{AC}{AB}\).
Write the power of point \(A\) with respect to circle \((BCDE)\).
From point \(A\), two secants to circle \((BCDE)\) are drawn: \(ADB\) and \(AEC\). Therefore \(AD\cdot AB=AE\cdot AC\). Dividing both sides by \(AE\cdot AB\), we get \(\frac{AD}{AE}=\frac{AC}{AB}\).
A key pattern for problems with points on two sides of a triangle.