## Quick Reference

The following is a summary of some of the theorems that are concerned with properties of a circle:

Let *A* and *B* be two points on a circle with centre *O*. If *P* is any point on the circumference of the circle and on the same side of the chord *AB* as *O*, then ∠*AOB*=2∠*APB*. Hence the ‘angle at the circumference’ ∠*APB* is independent of the position of *P*.

If *Q* is a point on the circumference and lies on the other side of *AB* from *P*, then *∠**AQB*=180°−∠*APB*. Hence opposite angles of a cyclic quadrilateral add up to 180°.When *AB* is a diameter, the angle at the circumference is the ‘angle in a semicircle’ and is a right angle. If *T* is any point on the tangent at *A*, then ∠*APB*=∠*BAT*.

Suppose now that a circle and a point *P* are given. Let any line through *P* meet the circle at points *A* and *B*. Then *PA*. *PB* is constant; that is, the same for all such lines. If *P* lies outside the circle and a line through *P* touches the circle at the point *T*, then *PA*.*PB*=*PT*^{2}.

*Subjects:*
Mathematics.

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