Similar triangles in a circle

Geometry, grade 10 (5 units) Lesson 24 of 26Learning

Similar triangles in a circle

Two chords that meet

Inscribed angles on the same arc are equal, so a circle gives equal angles – and with two pairs of equal angles, similar triangles (AA). The example below is the first of three such results; you meet them as exercises, each proved by its own pair of similar triangles.

Example: the chords $\displaystyle AB$ and $\displaystyle CD$ meet at $\displaystyle E$ . Then $\displaystyle {AE\cdot EB}={CE\cdot ED}$ .

The proof below shows that the triangles $\displaystyle AEC$ and $\displaystyle DEB$ are similar, and multiplies out the proportion of their sides:

	Claim	Reason
1	$\displaystyle {\angle BAC}={\angle BDC}$	Inscribed angles on the same arc $\displaystyle BC$
2	$\displaystyle {\angle AEC}={\angle DEB}$	Vertical angles
3	$\displaystyle \triangle AEC\sim\triangle DEB$	(1), (2); the AA similarity theorem
4	$\displaystyle {\frac{AE}{DE}}={\frac{CE}{BE}}$	(3); corresponding sides of similar triangles are in proportion
5	$\displaystyle {AE\cdot BE}={CE\cdot DE}$	(4)

Exercise 1

The chords $\displaystyle AB$ and $\displaystyle CD$ meet at $\displaystyle E$ .
$\displaystyle {AE}={6}$ , $\displaystyle {EB}={4}$ and $\displaystyle {CE}={3}$ .
Find $\displaystyle ED$ .

Exercise 2

Two secants from a point $\displaystyle P$ outside a circle: one cuts it at $\displaystyle A$ and $\displaystyle B$ , the other at $\displaystyle C$ and $\displaystyle D$ ( $\displaystyle A$ between $\displaystyle P$ and $\displaystyle B$ , $\displaystyle C$ between $\displaystyle P$ and $\displaystyle D$ ).

Prove: $\displaystyle \triangle PAD\sim\triangle PCB$ .

Prove: $\displaystyle {PA\cdot PB}={PC\cdot PD}$ .

\displaystyle {PA}={3}

\displaystyle {AB}={5}

and

\displaystyle {PC}={4}

. Find

\displaystyle PD

Exercise 3

From a point $\displaystyle P$ outside a circle come a tangent $\displaystyle PT$ , touching it at $\displaystyle T$ , and a secant that cuts it at $\displaystyle A$ and $\displaystyle B$ ( $\displaystyle A$ between $\displaystyle P$ and $\displaystyle B$ ).

Prove: $\displaystyle \triangle PTA\sim\triangle PBT$ , and so $\displaystyle {PT^{2}}={PA\cdot PB}$ .
(A hint: the angle between the tangent and the chord $\displaystyle TA$ .)

\displaystyle {PA}={4}

and

\displaystyle {AB}={5}

. Find

\displaystyle PT

/ – 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

The radii of similar triangles’ circles

In similar triangles with similarity ratio $\displaystyle k$ , the radii of the circumscribed circles are in the ratio $\displaystyle k$ , and so are the radii of the inscribed circles:

\displaystyle \frac{R_1}{R_2}=\frac{r_1}{r_2}=k

The circumscribed circle: the centre $\displaystyle O$ and a side $\displaystyle BC$ make an isosceles triangle $\displaystyle BOC$ whose angle at $\displaystyle O$ is twice the angle $\displaystyle A$ (or its supplement, for an obtuse $\displaystyle A$ ). In similar triangles these isosceles triangles are similar too, and their bases $\displaystyle BC$ are in the ratio $\displaystyle k$ – so their legs, the radii, are as well.
The inscribed circle: its centre $\displaystyle I$ and a side $\displaystyle BC$ make a triangle $\displaystyle BIC$ whose angles at $\displaystyle B$ and at $\displaystyle C$ are half the triangle’s angles. These triangles are similar, with bases in the ratio $\displaystyle k$ – so their heights, the radii $\displaystyle r$ , are in the ratio $\displaystyle k$ .

Exercise 4

The sides of the triangle $\displaystyle ABC$ are $\displaystyle 6$ , $\displaystyle 8$ and $\displaystyle 10$ .
The triangle $\displaystyle DEF$ is similar to it, and its longest side is $\displaystyle 15$ .

\displaystyle ABC

is right-angled. Find the radius of its circumscribed circle. (A hint: the angle on a diameter.)

/ – 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 radius of the circumscribed circle of

\displaystyle DEF

/ – 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

The radius of the inscribed circle of

\displaystyle ABC

\displaystyle 2

. Find that of

\displaystyle DEF

/ – 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 5

$\displaystyle AB$ is a diameter of the circle, and $\displaystyle C$ is on the circle.
$\displaystyle CD\perp AB$ , and $\displaystyle D$ is on $\displaystyle AB$ .
Prove: $\displaystyle {CD\cdot CD}={AD\cdot DB}$ .

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 $\displaystyle P$
$\displaystyle PT^{2}$ $\displaystyle =$ $\displaystyle PA\cdot PB$