Concyclic points and cyclic quadrilaterals

Points that lie on one circle are concyclic. "When do four points lie on a circle?"

Theorem 10: if a segment AB subtends equal angles at two points C and D on the same side of AB, then A, B, C and D are concyclic. The circle through A, B, C exists (Theorem 1). If D were outside it, or inside it, an exterior angle would have to be strictly bigger than an angle equal to it, which is impossible. So D is on the circle.

A quadrilateral (4-gon) whose four corners are concyclic is a cyclic quadrilateral.

Theorem 11: opposite angles of a cyclic quadrilateral add to 180°.

• ∠BAD is half the angle that arc BCD subtends at the centre;
• ∠BCD is half the angle that arc BAD subtends at the centre;
• the two arcs together go all the way round: 360°;
• so ∠BAD + ∠BCD = ½ × 360° = 180°.

Theorem 12, the converse: if the opposite angles of a quadrilateral add to 180°, it is cyclic.

A quick consequence: the exterior angle at any corner of a cyclic quadrilateral equals the interior opposite angle, because both are 180° minus the same angle.