Problem
GEO-B1-M04-P019 Equal Angles on One Segment
#19
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\) and \(E\) lie on sides \(AB\) and \(AC\), respectively. It is known that \(\angle CDE=\angle CBE\). Prove that points \(B,C,D,E\) lie on one circle.
Both angles stand on the same segment \(CE\).
Angle \(\angle CDE\) is formed by rays \(DC\) and \(DE\), and angle \(\angle CBE\) by rays \(BC\) and \(BE\). Both angles stand on the same segment \(CE\). If two points \(B\) and \(D\) see segment \(CE\) under equal angles, then they lie on one circle with \(C\) and \(E\). Therefore \(B,C,D,E\) lie on one circle.
This is a preview of the inscribed angle theorem; if needed, treat it as a preparatory lemma.