A circle inscribed in a triangle

Geometry, grade 11 (4 units) Lesson 17 of 20Learning

A circle inscribed in a triangle

From grade 10: every point on an angle bisector is equally far from the angle’s arms; and a point inside an angle, equally far from its arms, is on its bisector. A point’s distance from a line is the length of the perpendicular from it to the line.

A circle inscribed in a triangle is a circle that touches all three sides of the triangle. The triangle is circumscribed about the circle.

The centre of such a circle is equally far from the three sides – each distance is a radius, perpendicular to the side at the point of contact.
So the centre is on the bisector of each of the triangle’s angles.

The centre of the inscribed circle

Theorem: The three angle bisectors of a triangle meet at one point, the centre of the circle inscribed in the triangle. So every triangle has an inscribed circle.

Given: $\displaystyle AI$ bisects $\displaystyle \angle A$ , and $\displaystyle BI$ bisects $\displaystyle \angle B$ .
To prove: $\displaystyle CI$ bisects $\displaystyle \angle C$ , and the circle with centre $\displaystyle I$ touches the three sides.

The proof below drops perpendiculars from $\displaystyle I$ to the three sides and shows that they are equal:

	Claim	Reason
1	$\displaystyle {IF}={IE}$	$\displaystyle I$ is on the bisector of the angle $\displaystyle A$ , so it is equally far from the angle’s arms
2	$\displaystyle {IF}={ID}$	$\displaystyle I$ is on the bisector of the angle $\displaystyle B$
3	$\displaystyle {ID}={IE}$	(1), (2)
4	$\displaystyle {\angle ICA}={\angle ICB}$	(3); $\displaystyle I$ is inside the angle $\displaystyle C$ and equally far from its arms, so it is on its bisector
5	$\displaystyle {ID}={IE}={IF}$	(1), (2); the circle with centre $\displaystyle I$ and radius $\displaystyle ID$ passes through $\displaystyle D$ , $\displaystyle E$ , $\displaystyle F$ , and each side is perpendicular to a radius at its end – so it is a tangent to the circle

The tangent segments from the vertices

The inscribed circle touches the sides $\displaystyle AB$ , $\displaystyle BC$ , $\displaystyle CA$ at $\displaystyle F$ , $\displaystyle D$ , $\displaystyle E$ . Two tangents leave each vertex, so there are two equal tangent segments:

$\displaystyle {AF}={AE}$ , $\displaystyle {BF}={BD}$ , $\displaystyle {CD}={CE}$ .
Each side is the sum of two such segments – for example $\displaystyle {AB}={AF+FB}$ .

Example: $\displaystyle {AB}={9}$ , $\displaystyle {BC}={8}$ , $\displaystyle {CA}={7}$ . We write $\displaystyle {AF}={AE}={x}$ ; then $\displaystyle {BF}={BD}={9-x}$ and $\displaystyle {CE}={CD}={7-x}$ . The side $\displaystyle BC$ is $\displaystyle BD+DC$ :

(9-x)+(7-x)=8\quad\Rightarrow\quad 16-2x=8\quad\Rightarrow\quad x=4

The inscribed circle of a right-angled triangle

In a right-angled triangle with legs $\displaystyle a$ , $\displaystyle b$ and hypotenuse $\displaystyle c$ , the radius $\displaystyle r$ of the inscribed circle has a simple formula:

Next to the right angle there is a square: two of its sides are radii, and two are the tangent segments from the right-angled vertex. So these two tangent segments equal $\displaystyle r$ .
The rest of the legs, $\displaystyle a-r$ and $\displaystyle b-r$ , are tangent segments too, and the hypotenuse is made of the two tangent segments from the acute vertices: $\displaystyle {(a-r)+(b-r)}={c}$ .

\displaystyle r

\displaystyle =

\displaystyle \frac{a+b-c}{2}

Example: in the triangle with sides $\displaystyle 3$ , $\displaystyle 4$ , $\displaystyle 5$ : $\displaystyle {r}={\frac{3+4-5}{2}}={1}$ .

Exercise 1

The circle inscribed in the triangle $\displaystyle ABC$ touches the sides $\displaystyle AB$ , $\displaystyle BC$ , $\displaystyle CA$ at $\displaystyle F$ , $\displaystyle D$ , $\displaystyle E$ .
$\displaystyle {AB}={12}$ , $\displaystyle {BC}={10}$ , $\displaystyle {CA}={8}$ .
Find $\displaystyle AF$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

The legs of a right-angled triangle are $\displaystyle 5$ and $\displaystyle 12$ .
Find the radius of the circle inscribed in the triangle.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The triangle $\displaystyle ABO$ is right-angled, and the point $\displaystyle O$ is the origin.
The side $\displaystyle AO$ is on the $\displaystyle x$ -axis, and the side $\displaystyle BO$ on the $\displaystyle y$ -axis.
A circle with centre $\displaystyle C$ , in the second quadrant, is inscribed in the triangle.
The sides $\displaystyle AO$ , $\displaystyle BO$ and $\displaystyle AB$ touch the circle at $\displaystyle D$ , $\displaystyle E$ and $\displaystyle F$ .

Prove: the quadrilateral $\displaystyle OECD$ is a square.

Prove: the quadrilateral $\displaystyle ADCF$ is a kite.

$\displaystyle D(-2,0)$ .
Find the circle’s equation.

\displaystyle D=(-2,0)

The slope of the side $\displaystyle AB$ is $\displaystyle \frac{3}{4}$ .

Find

\displaystyle \angle CAD

, to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find the area of the trapezium

\displaystyle ACEO

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: geometry, grade 11 (4 units)

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The equation of a circle

The canonical equation – the centre at the origin, the radius $\displaystyle R$
$\displaystyle x^{2}+y^{2}=R^{2}$
The general equation – the centre $\displaystyle (a,b)$ , the radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$
The distance between two points
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
The midpoint of a segment
$\displaystyle x_M=\frac{x_1+x_2}{2},\ y_M=\frac{y_1+y_2}{2}$
Perpendicular lines – $\displaystyle m_1$ , $\displaystyle m_2$ their slopes
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$

Angles and chords in a circle

An inscribed angle – half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
A quadrilateral inscribed in a circle – a pair of opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
A chord $\displaystyle AB$ – $\displaystyle d$ its distance from the centre
$\displaystyle \left(\frac{AB}{2}\right)^{2}+d^{2}$ $\displaystyle =$ $\displaystyle R^{2}$

Tangents

Two tangents from a point $\displaystyle P$ – $\displaystyle A$ , $\displaystyle B$ the points of contact
$\displaystyle PA=PB$
The tangent segment $\displaystyle PA$ – $\displaystyle O$ the centre
$\displaystyle PA^{2}+R^{2}$ $\displaystyle =$ $\displaystyle PO^{2}$

Trigonometry, circumference and area

The law of sines – $\displaystyle R$ the radius of the circle through the triangle’s vertices
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The area of a triangle – $\displaystyle \gamma$ the angle between the sides $\displaystyle a$ and $\displaystyle b$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$
The area of a parallelogram – $\displaystyle a$ , $\displaystyle b$ adjacent sides, $\displaystyle \alpha$ the angle between them
$\displaystyle A$ $\displaystyle =$ $\displaystyle ab\sin\alpha$
The sine of an obtuse angle
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The circumference
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi R$
The area of a disc
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi R^{2}$