Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2012-11-22 14:00:35

Agnishom
Real Member

Online

### 4 kissing circles

If the radii of each of the outer circles is r then show that radius of the inner circle is

Last edited by Agnishom (2012-11-22 14:03:47)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #2 2012-11-22 20:03:24

bob bundy
Moderator

Online

### Re: 4 kissing circles

hi Agnishom

See diagram.  Let the radius of the little circle be R

AB = BC = r

AC = R + r (why?)

So you should be able to use Pythagoras here.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #3 2012-11-23 00:17:48

Agnishom
Real Member

Online

### Re: 4 kissing circles

WOW! Thats a great proof

But what if the examiner asks me How do I know that BC = r (apart from intution)?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #4 2012-11-23 00:28:51

bob bundy
Moderator

Online

### Re: 4 kissing circles

OK.  Another diagram below.

This one shows a circle in the middle (B) and one on the left that crosses it (A) and one on the right that just touches it (C) .

The tangent to B where it crosses A cuts A again.

The tangent to B where it touches C is also a tangent for C.

General rule.  When two circles touch they share a common tangent.

A tangent is always at right angles to the radius at the point where the tangent touches.

So back to the previous diagram:

ABC = 90.  If you give a letter to a fourth point, you'll complete a square.  So BC = r

And AD is perpendicular to the common tangent at D (haven't drawn this line) and DC is too.

So ADC is a straight line.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #5 2012-11-23 00:36:46

Agnishom
Real Member

Online

### Re: 4 kissing circles

I see
Thanks
However I will like to ask you again if I have any doubts

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #6 2012-11-23 00:38:45

bob bundy
Moderator

Online

### Re: 4 kissing circles

Certainly.  Geometry and I have a long and happy relationship.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #7 2013-01-23 03:15:01

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

Did you steal my idea?

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!

## #8 2013-01-23 03:49:24

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

Keep this comment up; I'm linking it to the GeoGebra Forum.
There are:
4 Circles
22 Squares
24 Triangles
66 Rectangles
72 Irregular

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!

## #9 2013-01-26 14:06:27

Agnishom
Real Member

Online

### Re: 4 kissing circles

Hi n872yt3r.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #10 2013-01-26 22:26:12

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

I'm thinking you stole my idea because I did the same thing on a thread of 3 kissing circles with bobbym.
And the 4 circles etc. is for my upload on the GeoGebra Forum and on this site where you can upload GeoGebra programs.

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!
The content of the passage is a discussion between two users, Agnishom and bob bundy, on a math forum about a geometry problem involving four circles that are tangent to each other (kissing circles). The problem is to find the radius of the inner circle given the radii of the outer circles.

Here is the analysis:
single_question_with_answer

1. the passage contains a mathematical question: "if the radii of each of the outer circles is r then show that radius of the inner circle is \(\frac{r}{2 + \sqrt{3}}\)."
2. an answer to this question is provided by bob bundy, who gives a geometric proof using pythagoras' theorem and properties of tangents and radii.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2012-11-22 14:00:35

Agnishom
Real Member

Online

### 4 kissing circles

If the radii of each of the outer circles is r then show that radius of the inner circle is

Last edited by Agnishom (2012-11-22 14:03:47)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #2 2012-11-22 20:03:24

bob bundy
Moderator

Online

### Re: 4 kissing circles

hi Agnishom

See diagram.  Let the radius of the little circle be R

AB = BC = r

AC = R + r (why?)

So you should be able to use Pythagoras here.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #3 2012-11-23 00:17:48

Agnishom
Real Member

Online

### Re: 4 kissing circles

WOW! Thats a great proof

But what if the examiner asks me How do I know that BC = r (apart from intution)?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #4 2012-11-23 00:28:51

bob bundy
Moderator

Online

### Re: 4 kissing circles

OK.  Another diagram below.

This one shows a circle in the middle (B) and one on the left that crosses it (A) and one on the right that just touches it (C) .

The tangent to B where it crosses A cuts A again.

The tangent to B where it touches C is also a tangent for C.

General rule.  When two circles touch they share a common tangent.

A tangent is always at right angles to the radius at the point where the tangent touches.

So back to the previous diagram:

ABC = 90.  If you give a letter to a fourth point, you'll complete a square.  So BC = r

And AD is perpendicular to the common tangent at D (haven't drawn this line) and DC is too.

So ADC is a straight line.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #5 2012-11-23 00:36:46

Agnishom
Real Member

Online

### Re: 4 kissing circles

I see
Thanks
However I will like to ask you again if I have any doubts

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #6 2012-11-23 00:38:45

bob bundy
Moderator

Online

### Re: 4 kissing circles

Certainly.  Geometry and I have a long and happy relationship.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #7 2013-01-23 03:15:01

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

Did you steal my idea?

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!

## #8 2013-01-23 03:49:24

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

Keep this comment up; I'm linking it to the GeoGebra Forum.
There are:
4 Circles
22 Squares
24 Triangles
66 Rectangles
72 Irregular

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!

## #9 2013-01-26 14:06:27

Agnishom
Real Member

Online

### Re: 4 kissing circles

Hi n872yt3r.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #10 2013-01-26 22:26:12

n872yt3r
Power Member

Offline

### Re: 4 kissing circles

I'm thinking you stole my idea because I did the same thing on a thread of 3 kissing circles with bobbym.
And the 4 circles etc. is for my upload on the GeoGebra Forum and on this site where you can upload GeoGebra programs.

- n872yt3r
Math Is Fun Rocks!
By the power of the exponent, I square and cube you!
user
The content of the passage is a discussion between two users, Agnishom and bob bundy, on a math forum about a geometry problem involving four circles that are tangent to each other (kissing circles). The problem is to find the radius of the inner circle given the radii of the outer circles.

Here is the analysis:

TYPE: SINGLE_QUESTION_WITH_ANSWER

1. The passage contains a mathematical question: "If the radii of each of the outer circles is r then show that radius of the inner circle is \(\frac{r}{2 + \sqrt{3}}\)."
2. An answer to this question is provided by bob bundy, who gives a geometric proof using Pythagoras' theorem and properties of tangents and radii.