If a line is a tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle
If two segments from the same exterior point are tangent to a circle, then they are congruent
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center
If an angle is inscribed in a circle, then its measure half the measure of the intercepted arc
If two inscribed angles of a circle intercept the same arc, then the angles are congruent
An angle inscribed in a semicircle is a right angle
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs
If two chords intersect in the interior of the circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord
If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment
If a secant segment and a tangent segment share an endpoint outside of a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.