# Fermat Points?

Imgur

$$\triangle ABC$$ is rotated about point $$A$$ $$60^{\circ}$$ clockwise to form $$\triangle AB'C''$$. Then, $$\triangle ABC$$ is rotated about point $$B$$ $$60^{\circ}$$ clockwise to form $$\triangle A''BC'$$. Finally, $$\triangle ABC$$ is rotated about point $$C$$ $$60^{\circ}$$ clockwise to form $$\triangle A'B''C$$.

Problem 1: Prove that: $AC'=BA'=CB'$ and $AB''=BC''=CA''$

Problem 2: Prove that $$AC',BA',CB'$$ are concurrent, and so are $$AB'',BC'',CA''$$.

Side Note I think I have, by making those problems up, just rediscovered the first and second Fermat Point. Now that I think about it, what I did is basically the same as attaching equilateral triangles.

If you extend my questions a little bit (the first question to be exact), you will find that they prove that the first Fermat point guarantees the shortest distance between the point and each of the triangle's vertices. You may also want to try that out. Explore! In fact, this question was inspired by me trying to find a generalized Pompeiu's Theorem.

Note by Daniel Liu
4 years, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 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 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Wow. I am so terrible with geometry proofs that I'm not even going to look at this. But how did you make the picture? I notice that everything is exactly aligned the way it's supposed to be. :O

- 4 years, 3 months ago

Asymptote Vector Graphics Language.

- 4 years, 3 months ago

Aha. Oh yeah that's right! Nevermind. Dude did you see my solution to your other proof problem?

- 4 years, 3 months ago