Along with angle measurements are the segments and lengths of a circle. Knowing the theorems can be useful in solving for chords in a circle.

**Chord: **A line segment whose endpoints are on a circle.

**Secant: **A line that intersects a circle in two points.

**Tangent: **A line, line segment, or ray that intersects a circle in exactly one point.

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

- GB = GE, so \(\overline{AC} \cong \overline{DF}\)

When two chords intersect in the interior of a circle, each chord is divided into two segments called segments of the chord.

*Theorem: *If two chords intersect a circle so that one chord is divided into segments of lengths a and b and the other chord into lengths c and d, the product of the segments of one chord is equal to the product of segments of the second chord.

- In the picture above, \(\color{blue}{ab} \color{black}{ = } \color{red}{cd}\).

A **secant** segment contains a chord of a circle and has exactly one endpoint outside the circle. The part of the secant segment outside the circle is called an external segment.

- Segment a is a tangent segment.
- Segment b is called an external segment.
- Segment b + c is the secant segment.

*Theorem: *If two secants are drawn from a common point outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

- Then a(a + b) = c(c + d)

*Theorem:* If two **tangent **segments are drawn from the same point outside the circle, then the segments are equal.

- If \(\overline{CD}\) and \(\overline{CB}\) are tangent segments, then CD = CB

*Theorem:* If a tangent and a secant are drawn from a common external point, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

- Then \(a(a + b) = c^2\)