Miscellaneous Circle Problems

In one sentence

Mixed circle problems all look different, but there are only a few weapons to choose from. When you read the problem, don't rush to calculate: first translate the conditions in the figure into one of these three kinds of invariant:

Equal angles: inscribed angles on the same arc, tangent–chord angles, exterior angles of a cyclic quadrilateral.
Equal products: two lines through a point meeting the circle give $P A \cdot P B = P C \cdot P D$ .
Equal lengths: radii, tangent lengths, the two halves of a chord cut by a perpendicular diameter.

When you see this, think of that

Step 1 / 6

You see a diameter
Think right angle: an inscribed angle on a diameter is $9 0^{\circ}$ . Conversely, when you see a right angle on a circle, look for a diameter.
You see a tangent
You see the midpoint or the length of a chord
You see two equal angles, or two right angles
You see a product or a ratio
No figure is given

Predict first

$A B$ is a diameter of a circle, chord $C D ⊥ A B$ at $H$ , $A H = 2$ and $H B = 8$ . How long is chord $C D$ ?

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

Stop 1 · One problem, three weapons

One problem, several solutions

$A B$ is a diameter of a circle, chord $C D ⊥ A B$ at $H$ , $A H = 2$ and $H B = 8$ . Find the length of $C D$ .

Equal products: intersecting chords

Chords $A B$ and $C D$ meet at $H$ , so $C H \cdot H D = A H \cdot H B = 16$ .

The diameter $A B$ is perpendicular to chord $C D$ , so by the perpendicular-diameter theorem $C H = H D$ . Hence $C H^{2} = 16$ , $C H = 4$ and $C D = 8$ .

CostOne line, once you see that H is where two chords cross.

Equal lengths: radius and distance to the chord

The diameter is $A B = 10$ , so the radius is $5$ , and the distance from the center $O$ to $H$ is $O H = 5 - 2 = 3$ .

In right triangle $O H C$ , $C H = 5^{2} - 3^{2} = 4$ , so $C D = 8$ .

CostYou need the radius first, but no power of a point at all.

Equal angles: the right angle on a diameter

Join $A C$ and $B C$ . $A B$ is a diameter, so $∠ A C B = 9 0^{\circ}$ .

$∠ A C H$ and $∠ C B H$ are both complementary to $∠ B C H$ , so they are equal. Then $△ A C H \sim △ C B H$ , which gives $C H^{2} = A H \cdot H B = 16$ , $C H = 4$ and $C D = 8$ .

CostOne similarity, and you pick up a useful result on the way.

Stop 2 · Angles: first sort out where the vertex is

The inscribed angle on chord

A B

has two values. What do the readings at

P

and

Q

add up to?

When the central angle is $12 0^{\circ}$ , the inscribed angle on the major arc is $6 0^{\circ}$ and on the minor arc it is $12 0^{\circ}$ . Whenever a problem talks about “the inscribed angle on a chord”, or doesn't say which arc the point is on, you must give both answers.

Stop 3 · Concyclic points: draw the invisible circle

Only two lights are left: one measures angles, the other lengths. Can one be on while the other is off?

The angle condition and the length condition say the same thing. When a problem gives you angles but asks for a product of lengths (or the other way round), the bridge in between is often “four concyclic points”.

Practice

Exercise 1Basic

A circle has radius $2$ , chord $A B = 23$ , and $C$ is a point on the circle other than $A$ and $B$ . What can $∠ A C B$ be, in degrees? (List every possibility, separated by commas.)

Exercise 2Advanced

$P T$ is tangent to a circle at $T$ , and $P T = 6$ . A secant through $P$ meets the circle at $A$ then $B$ , with $P A = 3$ . Find the length of chord $A B$ .

Exercise 3Advanced

Quadrilateral $A B C D$ is inscribed in a circle, and its diagonals $A C$ and $B D$ meet at $P$ . $P A = 2$ , $P C = 6$ , $P B = 3$ and $A B = 4$ . Find the length of $C D$ .

Exercise 4Challenge

Two circles meet at $A$ and $B$ . A line through $A$ meets the two circles again at $C$ and $D$ , and a line through $B$ meets them again at $E$ and $F$ ( $C$ and $E$ are on the first circle, $D$ and $F$ on the second; $A$ is between $C$ and $D$ , and $B$ is between $E$ and $F$ ). Prove that $C E ∥ D F$ .

In one sentence

When you see this, think of that

Stop 1 · One problem, three weapons

Equal products: intersecting chords

Equal lengths: radius and distance to the chord

Equal angles: the right angle on a diameter

Stop 2 · Angles: first sort out where the vertex is

Stop 3 · Concyclic points: draw the invisible circle

Practice

Mistake Museum · Exhibits from this stop

No figure given, only one figure drawn