Let A, B, C, D be non-collinear points with AB = CD and angle ABC = angle BCD.

Prove that there is a circle which passes through A, B, C and D.

Comment: This problem appeared before with the stronger assumption that all three segments AB, BC, and CD are equal. In that case the center of the inscribed circle and the circumscribed circle were the same. In this case, it need not be true.