The Power of a Point

Try this

Drag $A$ around the circle. $P A$ and $P B$ both change. What about their product?
The rectangle on the right has length $P A$ and width $P B$ . The green dashed square has the same area. Does the square ever move?
With $P$ outside the circle, click “Turn A to the point of tangency”. Now $A$ and $B$ coincide. What does the rectangle become?
Drag $P$ inside the circle and turn the line again. Is the product still constant?
Drag $P$ farther from the center, then closer. How does the product change? Compare with the $∣ O P^{2} - r^{2} ∣$ line.

However the line turns,

P A \cdot P B = O P^{2} - r^{2}

and the right-hand side depends only on where $P$ is. The three theorems in your textbook differ only in where $P$ is:

$P$ inside the circle: chords $A B$ and $C D$ meet at $P$ , and $P A \cdot P B = P C \cdot P D$ (the intersecting chords theorem).
$P$ outside the circle: two secants, and $P A \cdot P B = P C \cdot P D$ (the secant–secant theorem).
$P$ outside the circle, with one line turned into a tangent: $P T^{2} = P A \cdot P B$ (the tangent–secant theorem).

Predict first

$P$ is outside the circle, and a secant meets the circle at $A$ and $B$ with $P A = 2$ and $P B = 8$ . What is the length of the tangent from $P$ to the circle?

Choose an answer first; then you can see the result.

Why

Proof: PA · PB = PC · PD

Step 1 / 6

Two lines
Draw two lines through $P$ : one meets the circle at $A$ then $B$ , the other at $C$ then $D$ . We want to prove the two products are equal.

$P A \cdot P B = P C \cdot P D ?$
Join up two triangles
Find two pairs of equal angles
Write the proportion, cross-multiply
Same with P inside the circle
So what is the product?

The number $O P^{2} - r^{2}$ is called the power of point $P$ with respect to the circle. With $P$ outside the circle it is positive and equals the square of the tangent length; with $P$ on the circle it is $0$ ; with $P$ inside it is negative, and its absolute value is $P A \cdot P B$ . The next stop treats it as a quantity in its own right.

Practice

Exercise 1Basic

Two chords $A B$ and $C D$ of a circle meet at $P$ , with $P A = 3$ , $P B = 8$ and $P C = 4$ . Find $P D$ .

Exercise 2Basic

$P T$ is tangent to a circle at $T$ , and a secant through $P$ meets the circle at $A$ then $B$ , with $P A = 4$ and $A B = 5$ . Find $P T$ .

Exercise 3Advanced

Circle $O$ has radius $5$ , and point $P$ is inside it with $O P = 3$ . Chord $A B$ passes through $P$ , and $P A = 2$ . Find $P B$ .

Exercise 4Challenge

$P T$ is tangent to circle $O$ at $T$ , and $P T = 6$ . Line $P O$ meets the circle at $A$ and $B$ ( $A$ closer to $P$ ), and $P A = 4$ . Find the radius of the circle.

Try this

Why

Proof: PA · PB = PC · PD

Practice

Mistake Museum · Exhibits from this stop

Using the chord length in the secant theorem

Pairing up segments from different lines