Equilateral Triangle Tangency

After looking at the diagram of Extremes with Equilaterals, a question came to mind:

Given two concentric circles of radii r1r_1 and r2r_2, find a function f(r1,r2)f(r_1,r_2) that outputs the side length of the smallest equilateral triangle such that one side is tangent to one of the circle and another side is tangent to the other circle. The sides themselves must be touching the circle, not just the extensions of the sides.

Note by Daniel Liu
5 years, 4 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Oooh that is tough. But won't this note help people with the problem? :P

Finn Hulse - 5 years, 4 months ago

Log in to reply

No, it won't. In fact, I am pretty sure it is impossible in my original problem for this double-tangency situation to occur. This new problem was just inspired by the previous problem.

Daniel Liu - 5 years, 4 months ago

Log in to reply

How?

Finn Hulse - 5 years, 4 months ago

Log in to reply

@Finn Hulse Sorry, edited comment to be more clear ¨\ddot\smile

Daniel Liu - 5 years, 4 months ago

Log in to reply

My first thought is that such a configuration is not possible. I found an assumption, corrected for it, and only have solutions where r1=12r2 r_1 = \frac{1}{2} r_2 . I'd be interesting in seeing the general case.

Calvin Lin Staff - 5 years, 4 months ago

Log in to reply

The vertices of the equilateral triangle not necessarily have to lie on the circles.

Also you missed the case where r1=r2r_1=r_2 I think. (using your interpretation) or I'm missing something on what you are interpreting it as.

Daniel Liu - 5 years, 4 months ago

Log in to reply

Ah yes. I was misled by the reference to the other question.

Haha, I assumed that r1r2 r_1 \neq r_2 , so that's another assumption that needed to be corrected. Thanks!

Calvin Lin Staff - 5 years, 4 months ago

Log in to reply

I think that r2 = 0 and then the second circle is a point circle then the r1 will be altitude of the equilateral triangle then side of the equilateral triangle 2r1÷√3

Hitesh Sai - 5 years, 4 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...