Prove that 5 points lie on the same circle

Question

How do I prove that 5 points lie on the same circle? I know about the theorem that opposite angles in a quadrilateral are supplementary, but how does that help me prove that 5 points lie on the same circle? Can I break apart the irregular pentagon into two quadrilaterals to show it?

Answer 1

If two of the sets of four points are concylic, they share three points. Those three points define a unique circle (or straight line in a degenerate case) - so the other two points must lie on the same circle.

Answer 2

Actually, (AFAIK) you only need four points. Start by choosing any three of the points, and draw segments between them. Construct the perpendicular bisectors of those segments; they should meet at a single point: the circumcenter of the triangle defined by the initial three points. Then draw the circle centered at that point containing the initial three points, and check if the fourth (fifth, sixth, etc.) point lies on that circle. Since the circumcenter is a unique point of a triangle, one cannot include the fourth+ point in the circle (if it isn't already) without removing one or more of the initial three points from the circle.