Problem
GEO-B1-M05-P026 Equal Arcs in a Cyclic Quadrilateral
#26
★★★★☆ Level 4 of 5
In cyclic quadrilateral \(ABCD\), it is known that \(\angle ABD=\angle DBC\). Prove that \(AD=DC\).
These two angles stand on chords \(AD\) and \(DC\).
Angle \(\angle ABD\) stands on chord \(AD\), and angle \(\angle DBC\) stands on chord \(DC\). In one circle, equal inscribed angles stand on equal chords. Since \(\angle ABD=\angle DBC\), we get \(AD=DC\).
Useful as the converse form of the problem about equal chords and an angle bisector.