Problem
GEO-B1-M05-P014 A Cyclic Trapezoid
#14
★★★☆☆ Level 3 of 5
Quadrilateral \(ABCD\) is cyclic, and \(AB\parallel CD\). Prove that \(AD=BC\).
Try to prove equality of angles standing on chords \(AD\) and \(BC\).
Since \(AB\parallel CD\), we have \(\angle ABD=\angle BDC\). In the cyclic quadrilateral, angles \(\angle BDC\) and \(\angle BAC\) stand on chord \(BC\), so they are equal. Hence \(\angle ABD=\angle BAC\). The first angle stands on chord \(AD\), and the second on chord \(BC\). Equal inscribed angles stand on equal chords, so \(AD=BC\).
A good combination: parallelism plus equal inscribed angles.