Circles and Chords

Circles and Chords

A chord is a segment that joins two points of the circle.
A diameter is a chord that contains the center of the circle.

Theorems:

1. In a circle, a radius perpendicular to a chord bisects the chord.
   
2. In a circle, a radius that bisects a chord is perpendicular to the chord.
3. In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Proof of Theorem 1:

Theorem:
In a circle, or congruent circles, congruent chords are equidistant from the center.
(converse) In a circle, or congruent circles, chords equidistant from the center are congruent.

Theorem:
In a circle, or congruent circles, congruent chords have congruent arcs.
(converse) In a circle, or congruent circles, congruent arcs have congruent chords.

Theorem:
In a circle, parallel chords intercept congruent arcs.