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 noncollinear 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