Using Circle Properties

In one sentence

Stand on a circle and look at a chord: wherever you stand on the same arc, the angle you see is the same.

Inscribed angles on the same arc are equal: each is half the central angle.

This makes a circle a machine for moving angles: an angle can slide along the circle to somewhere else without changing size. So two angles that are far apart can be set equal straight away, as long as their vertices and two other points lie on one circle.

Even more important is using it backwards: the problem has no circle, but four points have “opposite angles supplementary” or “equal angles on the same side”, so they lie on a circle. Bring that invisible circle out, and the angles can move.

Predict first

The chord $A B$ has a central angle of $10 0^{\circ}$ . $P$ is on the circle, on the same side of $A B$ as the center. How many degrees is $∠ A P B$ ?

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

Stop 1 · An angle sliding along a circle (Vol. 5 · Angles in a circle)

$A$ and $B$ are two points on a circle, and the minor arc $A B$ is $13 0^{\circ}$ . $P$ moves around the circle. What can $∠ A P B$ be?

Example: looking at chord AB from the circle

Step 1 / 4

Measure it
$P$ is on the long arc at the top. The reading is $∠ A P B = 6 5^{\circ}$ .

$∠ A P B = 6 5^{\circ}$
Slide P along
Compare it with the central angle
Past an endpoint, to the other side

There are only two answers: $6 5^{\circ}$ or $11 5^{\circ}$ , and they are supplementary.

Stop 2 · Two right angles bring out a circle (Vol. 5 · Concyclic points)

In acute $△ A B C$ , $B E ⊥ A C$ at $E$ and $C F ⊥ A B$ at $F$ . Prove that $∠ A F E = ∠ A C B$ .

There is no circle in the figure. But there are two right angles, both standing on $B C$ .

Example: the feet of two altitudes

Step 1 / 4

Two angles far apart
$∠ A F E$ is at the top left and $∠ A C B$ at the bottom right. No parallel lines, no ready-made congruent or similar triangles. Yet when you drag $A$ , the two readings always agree.

$∠ A F E = ? ∠ A C B$
Two right angles on BC
Draw the circle
Opposite angles are supplementary

A bonus: $△ A F E$ and $△ A C B$ have two pairs of equal angles, so they are similar.

Stop 3 · Opposite angles supplementary: inside and outside (Vol. 4 · Cyclic and tangential quadrilaterals)

$A$ , $B$ , $C$ are on a circle. Where must a fourth point $D$ be for $∠ A + ∠ C = 18 0^{\circ}$ ?

Example: drag D in and out

Step 1 / 4

D on the circle
All four points are on the circle. $∠ A = 10 0^{\circ}$ and $∠ C = 8 0^{\circ}$ , which add up to $18 0^{\circ}$ .

$∠ A + ∠ C = 10 0^{\circ} + 8 0^{\circ} = 18 0^{\circ}$
Why
Drag D inside the circle
Drag D outside the circle

So “opposite angles supplementary” is both a property of cyclic quadrilaterals and a test for four concyclic points. Stop 2 used exactly this.

Stop 4 · A product that never changes (Vol. 5 · Power of a point: applications)

Two chords $A B$ and $C D$ of a circle meet at $P$ . $P A = 2$ , $P B = 6$ , $P C = 3$ . Find $P D$ .

Example: intersecting chords

Step 1 / 4

Four segments
$P$ splits each chord into two parts. Three parts are known; find the fourth.

$P A = 2, P B = 6, P C = 3, P D = ?$
Move angles with the circle
Write the proportion
Substitute

Practice

Exercise 1Basic

The chord $A B$ has a central angle of $14 0^{\circ}$ . $P$ is on the minor arc $A B$ (not at $A$ or $B$ ). How many degrees is $∠ A P B$ ?

Exercise 2Advanced

Two chords $A B$ and $C D$ of a circle meet at $P$ , with $P A = 4$ and $P B = 9$ , and $P$ is exactly the midpoint of $C D$ . Find the length of $C D$ .

Exercise 3Challenge

In acute $△ A B C$ , $∠ A = 5 0^{\circ}$ , and the altitudes $B E$ and $C F$ meet at $H$ . How many degrees is $∠ B H C$ ?

In one sentence

Stop 1 · An angle sliding along a circle (Vol. 5 · Angles in a circle)

Example: looking at chord AB from the circle

Stop 2 · Two right angles bring out a circle (Vol. 5 · Concyclic points)

Example: the feet of two altitudes

Stop 3 · Opposite angles supplementary: inside and outside (Vol. 4 · Cyclic and tangential quadrilaterals)

Example: drag D in and out

Stop 4 · A product that never changes (Vol. 5 · Power of a point: applications)

Example: intersecting chords

Practice

Mistake Museum · Exhibits from this stop

Using inscribed angles before checking the points are concyclic

Mixing up the two sides of a chord

Stops on this line