Problem
GEO-B2-M02-P014 The Converse Problem in a Triangle
#14
★★★★☆ Level 4 of 5
In triangle \(ABC\), points \(D\in AB\) and \(E\in AC\) satisfy \(AD\cdot AB=AE\cdot AC\). Prove that points \(B,C,D,E\) lie on one circle.
Apply the converse criterion to rays \(AB\) and \(AC\) starting at \(A\).
On ray \(AB\) lie points \(D,B\), and on ray \(AC\) lie points \(E,C\). The equality \(AD\cdot AB=AE\cdot AC\) has the form of equality of products of two secants from point \(A\). By the converse power criterion, points \(D,B,E,C\), that is \(B,C,D,E\), lie on one circle.
This is more important than a numerical problem: the student learns to prove cyclicity through a product.