Problem
GEO-B1-M07-P010 Draw a Circle From Equal Angles
#10
★★☆☆☆ Level 2 of 5
Points \(C\) and \(D\) lie on the same side of line \(AB\), and \(\angle ACB=\angle ADB\). Prove that \(A,B,C,D\) lie on one circle.
Draw the circle through \(A,B,C\) and check where point \(D\) must lie.
Draw the circle through \(A,B,C\). On one side of \(AB\), all points from which segment \(AB\) is seen under angle \(\angle ACB\) lie on one arc of this circle. Since \(\angle ADB=\angle ACB\), point \(D\) lies on this circle. Hence \(A,B,C,D\) are cyclic.
An important converse tool to equal inscribed angles.