Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point"). If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem). A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon.

If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd. Construct a circle through points A, B and C by finding the perpendicular bisectors (red) of the sides of the triangle (blue). Only two of the three bisectors are needed to find the centre. Construction through three noncollinear pointsIf two angles are inscribed on the same chord and on the same side of the chord, then they are equal. The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality).

Note: Secant is not a term you are required to know at GCSE, however it is important to note the difference between a chord and a secant. If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠ BOA and ∠ BPA are supplementary. Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the centre of the circle to which their arc belongs to. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. [16] A line segment going from one point of the circumference to another but does not go through the centre. The centre of the circle is the fixed point from which all points on the boundary of the circle are equidistant. Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.

Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one. Label the point of intersection of these two perpendicular bisectors M. (They meet because the points are not collinear).About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [17] If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠ DAQ = 1 / 2arc( AQ).