## Geometry puzzle

**Moderators:** jestingrabbit, Moderators General, Prelates

### Geometry puzzle

My friend gave me a puzzle today in which three circles with radius 1 are lined up, with the middle one intersecting each of the other two at a point, like so: OOO. They are inscribed in a semicircle so that the diameter intersects each circle along the bottom and the curved part intersects each of the end circles. My task is to find the radius of the big circle.

Sorry for the odd description; I attached a picture. It isn't perfect but you get the idea.

I'm sure you all will figure it out pretty quick. I just want to know: Can this be solved after taking High School Algebra 2 and Geometry? It'd be disappointed to find that after giving up the solution involved a math concept I simply wasn't familiar with.

Thanks, and happy geometry'ing!

Sorry for the odd description; I attached a picture. It isn't perfect but you get the idea.

I'm sure you all will figure it out pretty quick. I just want to know: Can this be solved after taking High School Algebra 2 and Geometry? It'd be disappointed to find that after giving up the solution involved a math concept I simply wasn't familiar with.

Thanks, and happy geometry'ing!

- captainwalrus
**Posts:**11**Joined:**Tue Nov 03, 2009 2:37 am UTC

### Re: Geometry puzzle

Hmmm, You can do it but I haven't taken geometry since 9th grade, so I don't really remember that well... Why wouldn't it just be 5 though?

It's over 9 x 10^3!

### Re: Geometry puzzle

I don't think it requires anything that you wouldn't know. Also, the answer isn't 5.

- phlip
- Restorer of Worlds
**Posts:**7543**Joined:**Sat Sep 23, 2006 3:56 am UTC**Location:**Australia-
**Contact:**

### Re: Geometry puzzle

I'm pretty sure it's

A hint, if you're trying to figure it out:

**Spoiler:**

A hint, if you're trying to figure it out:

**Spoiler:**

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};`

void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}

### Re: Geometry puzzle

I just looked at the hint.

I was thinking that it looked that way, but I wasn't sure how to prove it. So there's a postulate stating that? If that's the case, then it'd just be:

, right?

I was thinking that it looked that way, but I wasn't sure how to prove it. So there's a postulate stating that? If that's the case, then it'd just be:

**Spoiler:**

, right?

- Cosmologicon
**Posts:**1806**Joined:**Sat Nov 25, 2006 9:47 am UTC**Location:**Cambridge MA USA-
**Contact:**

### Re: Geometry puzzle

It's pretty easy to show, I thinkHertafeld wrote:I just looked at the hint.

I was thinking that it looked that way, but I wasn't sure how to prove it. So there's a postulate stating that?

**Spoiler:**

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

### Re: Geometry puzzle

Er, cosmo, the smaller circle is tangent to the larger circle on the interior, so that argument doesn't work as given. The same idea works though.

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

### Re: Geometry puzzle

so, if I understand this correctly, the diameter of the large circle is tangential to each of the three smaller circles, and the two outer small circles are both tangential to the large circle?

### Re: Geometry puzzle

1) Draw three circles of radius 1 along a horizontal line just touching each other (well, the two outer ones don't touch each other..)

2) Draw a horizontal line across the bottom

3) complete the semi-circle so that the arc just touches the two outer circles.

That's what it looks like.

2) Draw a horizontal line across the bottom

3) complete the semi-circle so that the arc just touches the two outer circles.

That's what it looks like.

- phlip
- Restorer of Worlds
**Posts:**7543**Joined:**Sat Sep 23, 2006 3:56 am UTC**Location:**Australia-
**Contact:**

### Re: Geometry puzzle

Poohblah wrote:so, if I understand this correctly, the diameter of the large circle is tangential to each of the three smaller circles, and the two outer small circles are both tangential to the large circle?

Yep:

Since I'm drawing it anyway, here's the full solution:

**Spoiler:**

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};`

void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}

### Re: Geometry puzzle

Yep, it certainly makes sense now. Thanks everyone!

**Spoiler:**

### Re: Geometry puzzle

Enjoy this one,

Find the radius given the following situation with a quadrant of a circle and a square of side length 1,

http://img291.imageshack.us/img291/2664/34353904.jpg

Find the original radius.

I accidentally exported the whole thing, not just the image, so it's very small, but the image is large (that's why I posted a direct link) and I didn't save it.

If you guys/girls want it bigger I'll do it again. The square is tangential to the appropriate sections of the quadrant.

Find the radius given the following situation with a quadrant of a circle and a square of side length 1,

http://img291.imageshack.us/img291/2664/34353904.jpg

Find the original radius.

I accidentally exported the whole thing, not just the image, so it's very small, but the image is large (that's why I posted a direct link) and I didn't save it.

If you guys/girls want it bigger I'll do it again. The square is tangential to the appropriate sections of the quadrant.

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

### Re: Geometry puzzle

It's not technically clear from that diagram that vertical reflection is a symmetry of your figure; without that additional assumption, the problem does not have a unique solution.

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

### Re: Geometry puzzle

As far as I know, there's only one way to have a square tangential to all sides, which implies the angles. If you don't accept this (or it isn't true) the triangle made by the square is a 45,45,90 triangle.

But you don't need to know this in order to solve the problem.

But you don't need to know this in order to solve the problem.

### Re: Geometry puzzle

Correct, but below is a much simpler way..

**Spoiler:**

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