Question

If the line segment joining two points subtends equal angles at two other points lying on the same side of the line, then all the four points (will/ may not) lie on a circle.

Answer 1

Consider the following example:
The measure of all angles subtended by an arc on the same segment of the circle are equal.

$\therefore \angle A=\angle D$

Conversely, it can be said that when the line segment CB joining two points C and B subtends equal angles at two other points D and A lying on the same side of the line, i.e., $\angle A=\angle D$, then all the four points A, B, C and D lie on a circle.