Problem
GEO-B2-M01-P002 The Converse Cyclicity Criterion
#2
★★☆☆☆ Level 2 of 5
Four distinct points \(A,B,C,D\) are such that no three of them are collinear and \(\angle ABC\equiv\angle ADC\pmod{180^\circ}\). Prove that \(A,B,C,D\) lie on one circle.
Consider the circle through \(A,B,C\) and recall the locus of points from which segment \(AC\) is seen under a fixed angle.
Draw the circle through \(A,B,C\). Every point of this circle, different from \(A\) and \(C\), sees chord \(AC\) under an angle equal to \(\angle ABC\) in oriented notation. By the condition, point \(D\) sees the same segment \(AC\) under the same oriented angle. Therefore \(D\) belongs to this circle, and points \(A,B,C,D\) are cyclic.
It is important to mention non-degeneracy: without it the criterion cannot be applied mechanically.