Triangles and quadrilaterals circumscribing a circle

Geometry, grade 10 (5 units) Lesson 23 of 26Learning

Triangles and quadrilaterals circumscribing a circle

The circle inscribed in a triangle

A polygon circumscribes a circle when each of its sides is a tangent to the circle; the circle is inscribed in it.

Every triangle circumscribes a circle: the angle bisectors meet at a point $\displaystyle I$ at equal distances from the three sides (the lesson The circumscribed circle, the inscribed circle and the altitudes).
The circle with centre $\displaystyle I$ through the feet $\displaystyle D$ , $\displaystyle E$ , $\displaystyle F$ of the perpendiculars from $\displaystyle I$ to the sides is perpendicular to each radius $\displaystyle ID$ , $\displaystyle IE$ , $\displaystyle IF$ at its end – so each side is a tangent.
From each vertex come two tangents, and they are equal: $\displaystyle {AF}={AE}$ , $\displaystyle {BF}={BD}$ , $\displaystyle {CD}={CE}$ .

Exercise 1

In the triangle in the drawing, $\displaystyle {AB}={7}$ , $\displaystyle {BC}={8}$ and $\displaystyle {AC}={9}$ .
The inscribed circle touches $\displaystyle AB$ at $\displaystyle F$ , $\displaystyle BC$ at $\displaystyle D$ and $\displaystyle AC$ at $\displaystyle E$ .

\displaystyle {AF}={AE}={x}

, so

\displaystyle {BF}={BD}={7-x}

and

\displaystyle {CE}={CD}={9-x}

. Find

\displaystyle x

from

\displaystyle {BC}={8}

Find $\displaystyle BD$ and $\displaystyle CD$ .
Show that $\displaystyle AF$ equals half the perimeter minus $\displaystyle BC$ .

\displaystyle BD

/ – 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

\displaystyle CD

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Exercise 2

A circle with centre $\displaystyle I$ and radius $\displaystyle r$ is inscribed in the right-angled triangle $\displaystyle ABC$ ( $\displaystyle {\angle C}={90^{\circ}}$ ).
It touches $\displaystyle BC$ at $\displaystyle D$ and $\displaystyle AC$ at $\displaystyle E$ .

Explain why $\displaystyle IDCE$ is a square, and so $\displaystyle {CD}={CE}={r}$ .

The legs are $\displaystyle a$ and $\displaystyle b$ , and the hypotenuse $\displaystyle c$ .
Use the tangents from $\displaystyle A$ and from $\displaystyle B$ to show that $\displaystyle {r}={\frac{a+b-c}{2}}$ .

The legs are

\displaystyle 6

and

\displaystyle 8

. Find

\displaystyle r

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A quadrilateral circumscribing a circle

Theorem: In a quadrilateral circumscribing a circle, the sum of two opposite sides equals the sum of the other two.

\displaystyle AB+CD

\displaystyle =

\displaystyle BC+AD

Given: the circle with centre $\displaystyle I$ touches the sides of $\displaystyle ABCD$ at $\displaystyle P$ , $\displaystyle Q$ , $\displaystyle R$ , $\displaystyle S$ .
To prove: $\displaystyle {AB+CD}={BC+AD}$ .

The proof below splits each side into two tangent segments, and pairs the equal ones:

	Claim	Reason
1	$\displaystyle {AP}={AS}$	Two tangents to a circle from the point $\displaystyle A$
2	$\displaystyle {BP}={BQ}$	Two tangents from $\displaystyle B$
3	$\displaystyle {CQ}={CR}$	Two tangents from $\displaystyle C$
4	$\displaystyle {DR}={DS}$	Two tangents from $\displaystyle D$
5	$\displaystyle {AB+CD}={AP+BP+CR+DR}$	$\displaystyle P$ is on $\displaystyle AB$ , and $\displaystyle R$ on $\displaystyle CD$
6	$\displaystyle {BC+AD}={BQ+CQ+AS+DS}$	$\displaystyle Q$ is on $\displaystyle BC$ , and $\displaystyle S$ on $\displaystyle AD$
7	$\displaystyle {AB+CD}={BC+AD}$	(1)–(6)

The converse theorem: a convex quadrilateral in which the sums of opposite sides are equal circumscribes a circle. Its proof is by contradiction, like that of the converse for an inscribed quadrilateral.

Every rhombus and every kite circumscribe a circle: $\displaystyle {a+a}={a+a}$ , and $\displaystyle {a+b}={b+a}$ .
A rectangle that is not a square does not: $\displaystyle a+a\neq b+b$ .

Exercise 3

The quadrilateral $\displaystyle ABCD$ circumscribes a circle.
$\displaystyle {AB}={2x}$ , $\displaystyle {BC}={9}$ , $\displaystyle {CD}={x+3}$ , $\displaystyle {DA}={12}$ .

Find

\displaystyle x

Find the perimeter of

\displaystyle ABCD

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Exercise 4

An isosceles trapezoid circumscribes a circle.
Its bases are $\displaystyle 4$ and $\displaystyle 16$ .

Find the length of a leg.

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Find the height of the trapezoid. (A hint: the perpendicular from an end of the short base cuts off

\displaystyle \frac{16-4}{2}

of the long one.)

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Find the radius of the circle. (A hint: the bases are parallel tangents.)

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Formulas: geometry, grade 10 (5 units)

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Introduction to analytic geometry

The distance between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}$
The midpoint of the segment
$\displaystyle M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)$
The slope of a line through two points
$\displaystyle m$ $\displaystyle =$ $\displaystyle \frac{y_2-y_1}{x_2-x_1}$ $\displaystyle x_1\neq x_2$
The line of slope $\displaystyle m$ through the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The explicit equation of a line – $\displaystyle m$ the slope, $\displaystyle b$ where it crosses the $\displaystyle y$ -axis
$\displaystyle y=mx+b$
The general equation of a line
$\displaystyle Ax+By+C=0$
Parallel lines
$\displaystyle m_1$ $\displaystyle =$ $\displaystyle m_2$
Perpendicular lines
$\displaystyle m_1\cdot m_2$ $\displaystyle =$ $\displaystyle -1$
A canonical circle – its centre at the origin, $\displaystyle R$ its radius
$\displaystyle x^{2}+y^{2}=R^{2}$
The circle with centre $\displaystyle (a,b)$ and radius $\displaystyle R$
$\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}$

Midsegments and the meeting point of the medians

A triangle’s midsegment – $\displaystyle DE$ is parallel to $\displaystyle BC$
$\displaystyle DE$ $\displaystyle =$ $\displaystyle \frac{1}{2}BC$
A trapezoid’s midsegment – $\displaystyle a$ , $\displaystyle b$ the bases
$\displaystyle EF$ $\displaystyle =$ $\displaystyle \frac{a+b}{2}$
The meeting point of the medians $\displaystyle G$ divides the median $\displaystyle AD$ in the ratio $\displaystyle 2:1$
$\displaystyle AG$ $\displaystyle =$ $\displaystyle 2GD$
Pythagoras’ theorem and its converse: the triangle is right-angled, the right angle opposite $\displaystyle c$
$\displaystyle a^{2}+b^{2}=c^{2}$

Proportion and similarity

A property of a proportion
$\displaystyle \frac{a}{b}=\frac{c}{d}\ \Rightarrow\ \frac{a+b}{b}=\frac{c+d}{d}$
Thales’ theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{DB}$ $\displaystyle =$ $\displaystyle \frac{AE}{EC}$
The extended Thales theorem – $\displaystyle DE\parallel BC$
$\displaystyle \frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$
The angle bisector theorem – $\displaystyle AD$ bisects the angle $\displaystyle A$
$\displaystyle \frac{BD}{DC}$ $\displaystyle =$ $\displaystyle \frac{AB}{AC}$
The similarity ratio – corresponding sides
$\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k$
Heights, medians, perimeters, the radii of the circumscribed and of the inscribed circles
$\displaystyle \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{P_1}{P_2}=\frac{R_1}{R_2}=\frac{r_1}{r_2}=k$
The ratio of the areas
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$

The circle

An inscribed angle is half the central angle on the same arc
$\displaystyle \angle ACB$ $\displaystyle =$ $\displaystyle \frac{1}{2}\angle AOB$
An inscribed angle on the diameter $\displaystyle AB$
$\displaystyle \angle ACB=90^{\circ}$
A quadrilateral $\displaystyle ABCD$ inscribed in a circle – opposite angles
$\displaystyle \angle A+\angle C$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The angle between the tangent $\displaystyle AT$ and the chord $\displaystyle AB$ equals an inscribed angle on $\displaystyle AB$ from its other side
$\displaystyle \angle BAT$ $\displaystyle =$ $\displaystyle \angle ACB$
Two tangents from the point $\displaystyle P$
$\displaystyle PA=PB$
A quadrilateral $\displaystyle ABCD$ circumscribing a circle
$\displaystyle AB+CD$ $\displaystyle =$ $\displaystyle BC+AD$
The chords $\displaystyle AB$ and $\displaystyle CD$ meet at the point $\displaystyle E$
$\displaystyle AE\cdot EB$ $\displaystyle =$ $\displaystyle CE\cdot ED$
Two secants from the point $\displaystyle P$
$\displaystyle PA\cdot PB$ $\displaystyle =$ $\displaystyle PC\cdot PD$
A tangent $\displaystyle PT$ and a secant $\displaystyle PAB$ from the point $\displaystyle P$
$\displaystyle PT^{2}$ $\displaystyle =$ $\displaystyle PA\cdot PB$