Course 12.3Lesson 3 of 3

Circles inside figures

A circle drawn inside a triangle or a four-sided figure touches every side. The equal-tangent theorem then gives a handful of neat rules: the tangent lengths from each corner of a triangle, the radius of the circle inside a right triangle, and the sums of opposite sides of a four-sided figure.

about 25 min1 simulation1 practice question1 Python program5 questions to check yourself

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The circle inside a triangle

Draw a triangle ABC. A circle can be drawn inside it so that it touches all three sides. It is called the incircle. Let it touch BC at D, CA at E and AB at F.

A right triangle ABC with the right angle at B in the lower left, A above B and C to the right. AB is 6 centimetres, BC is 8 centimetres and AC is 10 centimetres. A circle sits inside the triangle and touches all three sides, at F on AB, D on BC and E on AC. The radii from the centre to F, D and E are dashed and meet the sides at right angles. The tangent lengths are written along the sides: BF and BD are 2, AF and AE are 4, and CD and CE are 6. A line of text says the radius is 6 plus 8 minus 10, divided by 2, which is 2 centimetres.

Each corner is the outside point of two tangents. By the equal-tangent theorem,

\displaystyle AF = AE = x, \qquad BF = BD = y, \qquad CD = CE = z.

The picture shows this for a triangle with sides $6$ , $8$ and $10$ cm: $x = 4$ , $y = 2$ and $z = 6$ .

Each side of the triangle is made of two of these lengths:

\displaystyle AB = x + y, \qquad BC = y + z, \qquad CA = z + x.

Example. In triangle ABC, $AB = 9$ cm, $BC = 14$ cm and $CA = 11$ cm. Find the three tangent lengths.

Write the three sides. $x + y = 9$ , $y + z = 14$ and $z + x = 11$ .
Add them. $2(x + y + z) = 34$ , so $x + y + z = 17$ .
Subtract each side. $x = 17 - 14 = 3$ , $y = 17 - 11 = 6$ and $z = 17 - 9 = 8$ .

Check. $x + y = 9$ , $y + z = 14$ and $z + x = 11$ . All three agree.

The radius in a right triangle

Let the right angle be at B. Join the centre I to D and to F. The radii ID and IF are perpendicular to the sides BC and AB. So the figure BDIF has three right angles, and two equal adjacent sides $ID = IF = r$ . It is a square of side $r$ . Hence $BD = BF = r$ .

Now $AF = AB - r$ and $CD = BC - r$ . The hypotenuse is $AC = AE + EC = AF + CD$ . So

\displaystyle AC = AB + BC - 2r, \qquad \text{and so} \qquad r = \frac{AB + BC - AC}{2}.

Example. A right triangle has legs $6$ cm and $8$ cm. Its hypotenuse is $10$ cm. The radius of the incircle is $\dfrac{6 + 8 - 10}{2} = 2$ cm. For the triangle with sides $5$ , $12$ and $13$ it is $\dfrac{5 + 12 - 13}{2} = 2$ cm too.

A circle inside a four-sided figure

Now let a four-sided figure ABCD have a circle inside it that touches all four sides. Call the tangent lengths from A, B, C and D by $a$ , $b$ , $c$ and $d$ . Then

\displaystyle AB = a + b, \quad BC = b + c, \quad CD = c + d, \quad DA = d + a.

Add the first and third, and the second and fourth:

\displaystyle AB + CD = a + b + c + d = BC + DA.

The sums of opposite sides are equal. This gives two quick results.

The perimeter is $2(AB + CD)$ .
A parallelogram drawn round a circle must be a rhombus. Its opposite sides are equal, so $AB = CD$ and $BC = DA$ . Then $AB + CD = BC + DA$ gives $2AB = 2BC$ , so $AB = BC$ .

Example. A four-sided figure ABCD is drawn round a circle, with $AB = 7$ cm, $BC = 10$ cm and $CD = 12$ cm. Find DA.

Opposite sides. $AB + CD = BC + DA$ .
Substitute. $7 + 12 = 10 + DA$ , so $DA = 9$ cm.

Two circles with the same centre

Take two circles with the same centre O, and radii $R$ and $r$ , where $R > r$ . Let a chord AB of the larger circle touch the smaller circle at M.

OM is a radius of the smaller circle, and the chord is a tangent to it. So $OM \perp AB$ .
The perpendicular from the centre bisects a chord. So M is the midpoint of AB.
In the right triangle OMA, $AM^2 = R^2 - r^2$ .

Example. The radii are $5$ cm and $3$ cm. Then $AM = \sqrt{25 - 9} = 4$ cm, and the chord $AB = 8$ cm.

Example. A chord of the larger circle is $24$ cm long and touches a smaller circle of radius $5$ cm. Then $AM = 12$ cm and $R^2 = 144 + 25 = 169$ , so $R = 13$ cm.

One tangent to two circles

A line can touch two circles at once. It is a common tangent. Suppose the circles have radii $R$ and $r$ , with centres a distance $d$ apart, and the tangent does not cross the line between the centres. Draw the radii to the two points of contact. They are both perpendicular to the tangent, so they are parallel. A right triangle appears, with hypotenuse $d$ and one leg $R - r$ . The tangent segment between the points of contact is the other leg:

\displaystyle \text{length} = \sqrt{d^2 - (R - r)^2}.

For circles of radii $9$ cm and $4$ cm with centres $13$ cm apart, this is $\sqrt{169 - 25} = 12$ cm.

Practise with new numbers

Try it in Python

The program takes a triangle with sides $13$ , $14$ and $15$ cm. It finds the three tangent lengths from half the perimeter, checks that two of them make each side, and finds the radius of the incircle.

Common mistakes

Forgetting which lengths are equal. The two tangents from the same corner are equal, not the two on the same side.
Using the formula for any triangle. $r = \dfrac{a + b - c}{2}$ works only when the hypotenuse is $c$ .
Mixing up the sides in the four-sided figure. The equal sums are for opposite sides, $AB + CD$ and $BC + DA$ .
Taking half the chord as the radius. For two circles with one centre, half the chord is one leg, not a radius.

Remember these

Tangents from one corner are equal: $AF = AE$ , $BF = BD$ , $CD = CE$ .
Right triangle, incircle: $r = \dfrac{AB + BC - AC}{2}$ .
A four-sided figure round a circle: $AB + CD = BC + DA$ .
Two circles with one centre: half the chord is $\sqrt{R^2 - r^2}$ .

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

The incircle of triangle ABC touches AB at F, and $AF = 5$ cm. What is $AE$ , where E is the point of contact on AC?

In triangle ABC, $x + y = 7$ , $y + z = 9$ and $z + x = 8$ . Find $x + y + z$ .

A right triangle has legs $3$ cm and $4$ cm. What is the radius of its incircle?

A four-sided figure round a circle has $AB = 5$ cm, $CD = 8$ cm and $BC = 6$ cm. Find $DA$ .

Two circles share a centre, with radii $13$ cm and $12$ cm. A chord of the larger one touches the smaller. How long is it?

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Problems (4)

A circle inside triangle ABC touches AB at F, BC at D and CA at E…Medium · High School (9–10) · by StemPy Lab

A right triangle has legs 15 cm and 20 cm…Medium · High School (9–10) · by StemPy Lab

A four-sided figure ABCD is drawn round a circle, so that all four sides touch it…Medium · High School (9–10) · by StemPy Lab

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