Circles

Circle

The set of all points in a plane that are equidistant from a given point in the plane.

Radius

A segment whose endpoints are the center and any point on a circle

chord

A segment whose endpoints are on the circle

Diameter

A chord that contains the center of a circle

Secant

A line that intersects a circle at two points

Tangent

A line in the plane of a circle that intersects the circle in exactly one point.

Point of tangency

the point at which tangent passes through the circle

Tangent circles

coplanar circles that intersect at one point

Concentric Circles

Coplanar circles that have a common center

Common Tangent

A line or segment that is tangent or two coplanar circles

Tangent Line to Circle Theorem

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

Tangent Segments Theorem

Tangent segments from a common external point are congruent.

Central Angle

An angle whose vertex is the center of the circle

Semicircle

An arc with endpoints that are the endpoint of a diameter

Minor Arc

An arc with a measure less than 180 degrees, matching the measure of the central angle.

Major Arc

An arc with a measure more than 180 degrees, which is the difference of 360 degrees and the central angle.

Arc addition postulate

the measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Congruent circles

Circles in which a rigid motion or a composition of rigid motions maps one circle onto another.

Congruent Circles Theorem

Two circles are congruent if and only if they have the same radius or congruent radii.

Congruent Arcs

Arcs with the same measure if they are in the same circle or congruent circles

Congruent Central Angles Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Similar Arcs

Arcs with the same measure

Similar Circles Theorem

All circles are similar

Inscribed Angle

An angle whose <ABC vertex is on a circle and whose sides contain chords of the circle

Intercepted Arc

The arc that lies between two lines, rays, or segments (if the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to SUBTEND the angle)

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one-half the measure of its intercepted arc.

Inscribed Angles of a Circle Theorem

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Inscribed Polygons

A polygon inside a circle whose vertices all touch the edges of the circle.

Circumscribed Circle

The circle that contains the vertices of the polygon

Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is a right angle.

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Chord

A segment whose endpoints are on the circle

Congruent Corresponding Chords Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicular Chord Bisector Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Perpendicular Chord Bisector Theorem Converse

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter.

Equidistant Chords Theorem

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.