Math Core

When chords, secants and tangents cross, they chop each other into pieces. The lengths of those pieces are not random: in every case, a certain product of lengths on one line equals the same product on the other line. Three short equations cover all the cases, and one more rule handles the angles formed outside the circle.

Two chords crossing inside

When chords $\overline{AB}$ and $\overline{CD}$ intersect at a point $P$ inside a circle, each chord is split into two pieces.

Chords AB and CD cross at P, so PA · PB = PC · PD.

Chord–chord product

If two chords intersect inside a circle at $P$ , then

$PA \cdot PB = PC \cdot PD.$

The two pieces of one chord multiply to the same number as the two pieces of the other chord.

Why it works. Draw $\overline{AD}$ and $\overline{CB}$ . Angles $\angle A$ and $\angle C$ both intercept arc $\overset{\frown}{DB}$ , so they are congruent. The vertical angles at $P$ are congruent too. So $\triangle PAD \sim \triangle PCB$ by AA, and corresponding sides are proportional: $\dfrac{PA}{PC} = \dfrac{PD}{PB}$ . Cross-multiplying gives $PA \cdot PB = PC \cdot PD$ .

Worked example: Crossing chords

Chords $\overline{AB}$ and $\overline{CD}$ meet at $P$ . If $PA = 3$ , $PB = 8$ and $PC = 4$ , find $PD$ .

$4 \cdot PD = 3 \cdot 8 = 24 \quad\Longrightarrow\quad PD = 6.$

Two secants from outside

A secant segment starts at a point $P$ outside the circle and ends at the far intersection with the circle. The piece from $P$ to the near intersection is the external segment.

Secants PAB and PCD from point P. The near points are A and C; the far points are B and D.

Secant–secant product

If two secants are drawn from an outside point $P$ , then

$PA \cdot PB = PC \cdot PD,$

where $A$ and $C$ are the near points and $B$ and $D$ are the far points. In words: outside piece times whole secant is the same for both secants.

Worked example: Two secants

From point $P$ , one secant meets a circle at $A$ and $B$ with $PA = 4$ and $AB = 5$ . Another meets it at $C$ and $D$ with $PC = 3$ . Find $CD$ .

First find the whole first secant: $PB = 4 + 5 = 9$ . Then

$3 \cdot PD = 4 \cdot 9 = 36 \quad\Longrightarrow\quad PD = 12.$

So $CD = PD - PC = 12 - 3 = 9$ .

Common mistake

Use outside piece times whole secant, not outside piece times inside piece. In the example, the product is $4 \cdot 9$ , not $4 \cdot 5$ . If a problem gives you the chord inside the circle, add it to the outside piece first.

A tangent and a secant

If one of the two lines is a tangent, its "near point" and "far point" are the same point $T$ . So the tangent segment gets multiplied by itself.

Tangent PT and secant PAB. Here PA = 2, PB = 8 and PT = 4, and 4² = 2 · 8.

Tangent–secant product

If a tangent segment $\overline{PT}$ and a secant $PAB$ are drawn from the same outside point $P$ , then

$PT^2 = PA \cdot PB.$

Worked example: Finding a tangent length

From point $P$ , a secant meets a circle at $A$ and $B$ with $PA = 4$ and $PB = 16$ . Find the length of a tangent segment from $P$ .

$PT^2 = 4 \cdot 16 = 64 \quad\Longrightarrow\quad PT = 8.$

Angles outside the circle

Secants and tangents also form angles at the outside point $P$ . Such an angle intercepts two arcs: a far arc and a near arc.

Angle formed outside a circle

If two secants, two tangents, or a secant and a tangent meet at a point outside a circle, the angle they form is half the difference of the intercepted arcs:

$m\angle P = \dfrac{1}{2}\left(\text{far arc} - \text{near arc}\right).$

Compare the four angle rules you now know: an angle at the center equals its arc, an angle inside the circle is half the sum of its arcs, an angle on the circle is half its arc, and an angle outside the circle is half the difference.

Worked example: An angle outside

Two secants from $P$ intercept a far arc of $130^\circ$ and a near arc of $40^\circ$ . Find $m\angle P$ .

$m\angle P = \dfrac{1}{2}(130^\circ - 40^\circ) = \dfrac{1}{2}(90^\circ) = 45^\circ.$

Tip

All three product rules say the same thing: for any line through $P$ that meets the circle, (distance from $P$ to the first point) $\times$ (distance from $P$ to the second point) is constant. Measure every length from $P$ , and you'll never mix up the formulas.

Practice

Practice 1

Chords $\overline{AB}$ and $\overline{CD}$ intersect at $P$ . If $PA = 6$ , $PB = 4$ and $PC = 3$ , what is $PD$ ?