Circle theorems

• Angle in a semicircle = 90°; the angle between a tangent and a radius = 90°.
• The angle at the centre is twice the angle at the circumference.
• Angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.
• The alternate segment theorem.
• Symmetry properties: equal chords are equidistant from the centre; the perpendicular bisector of a chord passes through the centre; tangents from an external point are equal in length.

A diameter makes a right angle

If AB is a diameter and C is any other point on the circle, angle ACB = 90°.

Another right angle

A tangent meets the radius drawn to the point of contact at 90°. This usually leads to Pythagoras or trigonometry.

Twice the angle at the circumference

Angles standing on the same arc: the one at the centre is double the one at the circumference. The arc can be the major arc, which makes the angle at the centre reflex.

Same arc, same angle

Angles at the circumference standing on the same arc (on the same side of the chord) are equal. Look for a "bow tie" of two triangles sharing a chord.

Opposite angles are supplementary

A cyclic quadrilateral has all four vertices on the circle. Its opposite angles sum to 180°.

Tangent, chord and the opposite segment

The angle between a tangent and a chord at the point of contact equals the angle at the circumference in the alternate (other) segment.

Chords and tangents