# Thread: Smile, It's just math.

1. ## Smile, It's just math.

The diagram shows two equilateral triangles; the larger one has side 4.
The three larger circles are equal, touching each other and the sides of the larger triangle without overlapping.
The smaller triangle is constructed from the center points of the three larger circles.
The smiley face is a circle that touches all three sides of the inner triangle.
What is the diameter of the inner circle?

Disclaimer: the diagram is for illustrative purposes only. Under magnification, the circles may not appear to be round and the sides may not appear to be touching, but that's just too bad. All characters are fictional. No resemblance to actual persons was intended or implied. Your mileage may vary. Do not use near an open flame and do not try this at home. Well, okay, you can try this at home.

2. ## Re: Smile, It's just math.

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">2 - 2/3*SQRT(3)</span hide>

3. ## Re: Smile, It's just math.

Been working on this for several days... Decided to hang onto the prize until we can confirm your answer or <img src=/S/whisper.gif border=0 alt=whisper width=29 height=17> until you show us how you did it!
We got <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">8/9</span hide> and attached our work.

4. ## Re: Smile, It's just math.

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">NT = 2 and NU is 1/3 of NT, so NU = 2/3.
But that would mean that r = 2/3.
Since also s = 2/3, there is a contradiction here!</span hide>

5. ## Re: Smile, It's just math.

Thank for complicating MY PUZZLE even more <img src=/S/brickwall.gif border=0 alt=brickwall width=25 height=15>

We're stumped and running in circles. See latest attachment and see if you can save us. <img src=/S/confused.gif border=0 alt=confused width=15 height=20>

6. ## Re: Smile, It's just math.

I have attached your workbook with my solution.

7. ## Re: Smile, It's just math.

Neither of you seem to have addressed two issues:
1. There are 2 inner circles within the circle you are talking about.
2. Neither of those circles is attached to anything, so their diameters are unknowable.
<img src=/S/grin.gif border=0 alt=grin width=15 height=15>

8. ## Re: Smile, It's just math.

That's not <img src=/S/grin.gif border=0 alt=grin width=15 height=15> but <img src=/S/smile.gif border=0 alt=smile width=15 height=15>

9. ## Re: Smile, It's just math.

Good point! <img src=/S/thumbup.gif border=0 alt=thumbup width=15 height=15>

10. ## Re: Smile, It's just math.

I concur with you. But I can't see the inconsistency... yet.

Alan

11. ## Re: Smile, It's just math.

Ricky assumed that U is the intersection of the three angular bisectors of the equilateral blue triangle. This would imply that NU=2/3 and UT=4/3. But that isn't correct - as I show in the worksheet I attached, NU : Ut = 1 : SQRT(3), not 1 : 2.

12. ## Re: Smile, It's just math.

Ah yes - all is clear now.

Alan

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