Problem
GEO-B1-M05-P012 One Segment Under Equal Angles
#12
★★☆☆☆ Level 2 of 5
Points \(C\) and \(D\) lie on the same side of line \(AB\). It is known that \(\angle ACB=\angle ADB\). Prove that points \(A,B,C,D\) lie on one circle.
Points \(C\) and \(D\) see segment \(AB\) under the same angle.
The set of points from which segment \(AB\) is seen under a fixed angle lies on an arc of a circle through \(A\) and \(B\). Since \(C\) and \(D\) are on the same side of \(AB\) and \(\angle ACB=\angle ADB\), they lie on the same such arc. Therefore \(A,B,C,D\) lie on one circle.
This is the converse form of the main pattern \(\angle ABC=\angle ADC\).