Math Core

Draw any line through a point $P$ that hits a circle twice, at $A$ and $B$ . Surprisingly, the product $PA \cdot PB$ is the same for every such line. That single fact, called the Power of a Point, unlocks a huge number of AMC circle problems involving chords, secants and tangents.

The three cases

Power of a Point

Let $P$ be a point and $\omega$ a circle. For every line through $P$ meeting $\omega$ at $A$ and $B$ , the product $PA \cdot PB$ is the same. So:

Two chords meeting inside at $P$ : $PA \cdot PB = PC \cdot PD$ .
Two secants from an outside point $P$ : $PA \cdot PB = PC \cdot PD$ , where $A, C$ are the near points and $B, D$ the far points.
A tangent and a secant from $P$ : $PT^2 = PA \cdot PB$ , where $T$ is the point of tangency.

Two chords crossing at P (the dot). The two pieces of one chord have the same product as the two pieces of the other.

Why it works. For two chords $AB$ and $CD$ crossing at $P$ , the inscribed angles $\angle BAD$ and $\angle BCD$ subtend the same arc, so they are equal, and the vertical angles at $P$ are equal. So $\triangle PAD \sim \triangle PCB$ , which gives $\dfrac{PA}{PC} = \dfrac{PD}{PB}$ , or $PA \cdot PB = PC \cdot PD$ . The outside cases work the same way. A tangent is the limiting case of a secant whose two intersection points have merged into one.

The power as a number

Draw the line through $P$ and the center $O$ . It meets the circle at distances $d - r$ and $d + r$ from $P$ (where $d = OP$ and $r$ is the radius). So the common product is

$PA \cdot PB = |d^2 - r^2|.$

This is the power of $P$ . It is positive outside the circle (equal to the tangent length squared), zero on the circle, and negative inside if you use signed lengths.

Worked example: Crossing chords

Chords $AB$ and $CD$ meet at $P$ . If $AP = 4$ , $PB = 6$ and $CP = 3$ , find $PD$ .

$3 \cdot PD = 4 \cdot 6$ , so $PD = 8$ .

Worked example: Tangent and secant

From a point $P$ outside a circle, a secant meets the circle at $A$ and $B$ with $PA = 4$ and $AB = 12$ . Find the length of a tangent from $P$ .

$PB = 4 + 12 = 16$ , so $PT^2 = 4 \cdot 16 = 64$ and $PT = 8$ .

Common mistake

In the secant cases, the lengths are measured from $P$ : use $PA \cdot PB$ where $PB$ is the whole secant, not $PA \cdot AB$ . Convert "the chord has length $12$ " into $PB = PA + 12$ first.

Worked example: Finding a radius from a chord

A chord of a circle has length $24$ . The arc it cuts off rises $8$ above the chord at its midpoint. Find the radius.

Extend the perpendicular bisector of the chord to a full diameter. It crosses the chord at the midpoint $M$ and splits the diameter into pieces $8$ and $2r - 8$ . By crossing chords, $12 \cdot 12 = 8(2r - 8)$ , so $2r - 8 = 18$ and $r = 13$ .

Worked example: Using the distance to the center

Point $P$ is $3$ units from the center of a circle of radius $5$ . Every chord through $P$ is split by $P$ into two pieces. What is the product of the pieces, and what is the shortest chord through $P$ ?

The power is $r^2 - d^2 = 25 - 9 = 16$ , so every product is $16$ . The shortest chord is perpendicular to $OP$ , where the two pieces are equal: each is $4$ , so the chord has length $8$ .

Tip

Equal tangents. Two tangent segments from the same point to a circle have equal length. And if two circles meet at $A$ and $B$ , every point $P$ on line $AB$ has the same power $PA \cdot PB$ with respect to both circles, so the tangents from $P$ to the two circles are equal. That line is called the radical axis.

Practice

Practice 1

Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at $P$ . If $AP = 6$ , $PB = 8$ and $CD = 16$ , what is $|CP - PD|$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

From a point $P$ outside a circle, the tangent segment to the circle has length $10$ . A line through $P$ meets the circle at $A$ and $B$ , with $PA = 5$ . What is $AB$ ?

Practice 3

Point $P$ is $6$ units from the center of a circle of radius $10$ . What is the length of the shortest chord of the circle that passes through $P$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A circular arch spans $30$ feet at its base (the chord) and rises $5$ feet above the base at its highest point. What is the radius of the circle, in feet?

Practice 5

Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$ . Point $P$ lies on line $AB$ , outside both circles, with $PA = 4$ and $PB = 9$ . What is the length of a tangent segment from $P$ to $\omega_2$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

From a point $P$ outside a circle, one line meets the circle at $A$ and $B$ with $PA = 3$ and $AB = 9$ . A second line meets the circle at $C$ and $D$ (with $C$ between $P$ and $D$ ) so that $PC = CD$ . What is $PD$ ?

Practice 7

A circle has center $O$ and radius $5$ . Point $P$ is $13$ units from $O$ . A line through $P$ meets the circle at $A$ and $B$ with $AB = 8$ . What is $PA + PB$ ?

Practice 8

Circles of radius $3$ and $5$ are externally tangent at point $K$ . A common external tangent line touches the circles at $T_1$ and $T_2$ . The common tangent line through $K$ meets $\overline{T_1T_2}$ at $M$ . What is $MK$ ?

Real contest practice

2013 AMC 10A, Problem 23: a circle centered at a vertex cuts the opposite side; the power of a point gives an equation in integers.
2012 AMC 12A, Problem 16: two circles where one passes through the other's center; the power of a point finds the radius.