Definitions involving circles

Circle, center, radius: Let O be a point and let r be a positive real number. The circle C with center O and radius r is the set of all points P such that the distance from O to P equals r.

Chord, radius, diameter: A chord of a circle C is a segment with both endpoints on C. A radius of a circle C is a segment with one endpoint the center of C and the other endpoint on C. A diameter of C is a chord of C such that the center of C lies on the chord.

Tangent: The line L is tangent to the circle C at the point P if P lies on both L and C and the radius OP is perpendicular to L.

Arcs: An arc of a circle with endpoints P and Q is a major arc, a minor arc, or a semicircle. Let C be a circle with center O. Let P and Q be distinct points on C, such that the angle measure a(POQ) is not 180. The minor arc with endpoints P and Q is the set of all points X on C such that X = P or X = Q or Ray(OX) is between Ray(OP) and Ray(OQ). The major arc with endpoints P and Q is is the set of all points X on C such that X = P or X = Q or Ray(OX) is not between Ray(OP) and Ray(OQ). If a(POQ) = 180 then P and Q divide C into two semicircles. A semicircle of C determined by P and Q consists of the points P and Q and all the points which lie on one side of the line PQ.

Arc angle measure: Let C be a circle with center O. If A is a minor arc of C with endpoints P and Q, the arc angle measure of A is the angle measure of POQ. If A is a major arc of C with endpoints P and Q, the arc angle measure of A is 360 minus the angle measure of POQ. (For angle measure here we are using the GSP "degrees" measure, which is the same as "absolute angle measure" from the angle axiom.)

Inscribed angle: If the points P, Q, R are on the circle C, the angle PQR is inscribed in the circle C. The intercepted arc (also called the subtended arc) of the inscribed angle PQR is the set of all points X on C such that Ray(QX) is between Ray(QP) and Ray(QR).