Power of a Point: Applications

What this move does

The last stop found that $P A \cdot P B$ depends only on where $P$ is. Take this number, which depends only on position, on its own and call it the power of $P$ with respect to circle $O$ :

power = O P^{2} - r^{2}

With $P$ outside the circle the power is positive and equals the square of the tangent length. With $P$ on the circle it is $0$ . With $P$ inside the circle it is negative.

With it, problems about “two circles” or “three circles” follow a fixed three-step routine:

Write down the power of the key points with respect to each circle (a tangent length squared, or the product of the two pieces of a secant).
The points with equal power with respect to two circles all lie on one line, perpendicular to the line of centers. This line is the radical axis. When the circles intersect, the radical axis is the line through their common chord.
Three circles give a radical axis for each pair. The three radical axes either meet at one point (the radical center) or are parallel.

Predict first

Two circles meet at two points $A$ and $B$ . Which line is their radical axis?

The line of centers

O_{1} O_{2}

The line through the common chord

A B

The perpendicular bisector of

O_{1} O_{2}

A common tangent of the two circles

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

Work through one

Why the points of equal power form a line

Step 1 / 5

Pick any point
In general the powers of $P$ with respect to the two circles are different. We want every $P$ that makes them equal.

$P O_{1}^{2} - r_{1}^{2} = P O_{2}^{2} - r_{2}^{2} ?$
Rearrange
Drop a perpendicular, use Pythagoras
The foot is pinned down
That perpendicular is the radical axis

Three circles

Three circles have three radical axes. Say the first two meet at $P$ . $P$ has equal power with respect to circles $O_{1}$ and $O_{2}$ , and equal power with respect to circles $O_{2}$ and $O_{3}$ , so it also has equal power with respect to $O_{1}$ and $O_{3}$ : $P$ lands on the third radical axis automatically.

Drag the three centers or change the radii. Do the three radical axes ever stop meeting at one point?

This little “pass it along” argument is very useful: to prove three lines are concurrent, try to see them as the radical axes of three circles taken in pairs.

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: The three radical axes meet at one point

When the centers form a triangle, the three radical axes always share a point.

Practice

Exercise 1Basic

Circle $O$ has radius $3$ , and $O P = 5$ . Find the power of point $P$ with respect to circle $O$ .

Exercise 2Basic

Circle $O$ has radius $5$ , and point $P$ is inside it with $O P = 3$ . Find the power of point $P$ with respect to circle $O$ .

Exercise 3Advanced

Circle $O_{1}$ has radius $5$ , circle $O_{2}$ has radius $3$ , and $O_{1} O_{2} = 10$ . The radical axis of the two circles meets the line of centers at $H$ . Find the length $O_{1} H$ .

Exercise 4Challenge

Two circles meet at $A$ and $B$ . A common tangent touches the two circles at $S$ and $T$ . Line $A B$ meets segment $S T$ at $M$ . Prove that $M$ is the midpoint of $S T$ .

What this move does

Work through one

Why the points of equal power form a line

Three circles

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Taking the radical axis to be the perpendicular bisector of the centers

Writing the power of a point inside the circle as positive