# Circles

A circle is a locus of all points which are equidistant from a point. If O is a fixed point in a given plane, the set of points in the plane which are at equal distances from O will form a circle.

### Parts of Circle

Diameter is always twice the radius of the circle. The **secant** to the circle cut the circle at two different points. The **tangent **touches the circle at one and only one point.

### Properties of a Circle

- If two chords of a circle are equal, their corresponding arcs have equal measure.
- Measurement of an arc is the angle subtended at the centre. Equal arcs subtend equal angles at the center.
- A line from centre and perpendicular to a chord bisects the chord.
- Equal chords of a circle are equidistant from the centre.
- When two circles touch, their centres and their point of contact are collinear.
- If the two circles touch externally, the distance between their centres is equal to sum of their radii.
- If the two circles touch internally, the distance between the centres is equal to difference of their radii.
- Angle at the centre made by an arc is equal to twice the angle made by the arc at any point on the remaining part of the circumference.
- If two chords are equal then the arc containing the chords will also be equal.
- There can be one and only one circle that touches three non-collinear points.
- The angle inscribed in a semicircle is 90°.
- If two chords AB and CD intersect externally or internally at P, then PA × PB = PC × PD
- If PAB is a secant and PT is a tangent, then PT
^{2}= PA × PB

### Cyclic Quadrilateral

If a quadrilateral is inscribed in a circle i.e. all the vertex lies on the circumference of the circle, it is said to be cyclic quadrilateral.

- In a cyclic quadrilateral, opposite angles are supplementary.
- In a cyclic quadrilateral, if any one side is extended, the exterior angle so formed is equal to the interior opposite angle.

### Alternate Angle Theorem

Angles in the alternate segments are equal.

In the given figure, AC is a Chord touching the circle at points A and C. At point A we have a tangent PAT making ∠CAT and ∠CAP with the chord AC. In the circle ∠ABC and ∠ADC are two angles in two different segments.

Here for ∠CAT, the ∠ADC is in alternate segment and for ∠CAP; the ∠ABC is in alternate segment. So according to the statement of the theorem the pair of these alternate angles are equal to each other.

Then, ∠CAT = ∠ADC and ∠PAC = ∠ABC