Circle Geometry

Summary — Topic 19: Circle geometry

Angles in a circle

• Theorem 1: An angle with its vertex at the centre of the circle is twice the size of an angle subtended by the same arc but with the vertex at the circumference. Code [pic 2]
• Theorem 2: Angles that have their vertices on the circumference and are subtended by the same arc are equal. Code [pic 3]
• Theorem 3: Angles subtended by the diameter are right angles. Code [pic 4]
• Theorem 4: A tangent and a radius drawn to the same point on a circle meet at a 90° angle. Code [pic 5]
• Theorem 5: An angle formed by two tangents is bisected by the line joining the vertex of that angle to the centre of the circle. Code [pic 6]

Intersecting chords, secants and tangents

• Theorem 6: Code [pic 7]

[pic 8]

PX × QX = RX × SX

• Theorem 7: Code [pic 9]

[pic 10]

AX × XB = XC × DX

• Theorem 8: Code [pic 11]

[pic 12]

AC = BC

• Theorem 9: Code [pic 13]

[pic 14]

If OC ⊥ AB, AX = XB.

• Theorem 10: Code [pic 15]

[pic 16]

If MN = PR, then OD = OC.

Cyclic quadrilaterals

• A cyclic quadrilateral has all four of its vertices on the circumference of a circle.
• Theorem 11: Opposite angles of a cyclic quadrilateral are supplementary. Code [pic 17]
• Theorem 12: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. Code [pic 18]

Tangents, secants and chords

• Theorem 13: The angle formed by a tangent and a chord is equal to the angle in the alternate segment. Code [pic 19]

[pic 20]

∠BAD = ∠AED and ∠BAD = ∠AFD

• Theorem 14: If a tangent and a secant intersect as shown, then XA × XB = (XT)2. Code [pic 21]

[pic 22]