Problem
GEO-B1-M05-P019 An Angle Bisector Through Equal Chords
#19
★★★☆☆ Level 3 of 5
In cyclic quadrilateral \(ABCD\), it is known that \(BC=CD\). Prove that diagonal \(AC\) bisects angle \(BAD\).
Compare angles \(\angle BAC\) and \(\angle CAD\): which chords do they stand on?
Angle \(\angle BAC\) stands on chord \(BC\), and angle \(\angle CAD\) stands on chord \(CD\). By condition \(BC=CD\), so the corresponding arcs are equal. Therefore \(\angle BAC=\angle CAD\). Hence \(AC\) is the angle bisector of \(\angle BAD\).
A good olympiad micro-idea: equal chords can produce an angle bisector.