# Thread: Vertices of equilateral triangle

1. ## Vertices of equilateral triangle

If (0,0), (a,11), (b, 37) are the vertices of an equilateral triangle, find the product of a and b.

2. ## Re: Vertices of equilateral triangle

Originally Posted by shiny718
If (0,0), (a,11), (b, 37) are the vertices of an equilateral triangle, find the product of a and b.
Each side of the triangle is equal, so that means \displaystyle \displaystyle \begin{align*} \sqrt{(a - 0)^2 + (11 - 0)^2} = \sqrt{(b - a)^2 + (37 - 11)^2} = \sqrt{(b - 0)^2 + (37 - 0)^2} \end{align*}.

See if you can evaluate a and b from here.

3. ## Re: Vertices of equilateral triangle

Hi Prove it, I've tried evaluating the method u suggested above but I'm stuck with $\displaystyle a^2 -b^2 =1248, 2ab - b^2 =555$ and $\displaystyle a^2 -2ab =693$ Any tips on how to continue?

4. ## Re: Vertices of equilateral triangle

Originally Posted by shiny718
Hi Prove it, I've tried evaluating the method u suggested above but I'm stuck with $\displaystyle a^2 -b^2 =1248, 2ab - b^2 =555$ and $\displaystyle a^2 -2ab =693$ Any tips on how to continue?
From the last equation, write b in terms of a, then substitute into another equation.

5. ## Re: Vertices of equilateral triangle

This method will work, but in a competition situation, you'll have to be very good at bashing. A much simpler way to do this problem is to consider (b, 37) as a rotation of 60 degrees of (a, 11). This method is much easier and requires almost no bash. You can either use trigonometric sum identities, rotations on the complex plane, or rotation mapping of matrices; they are all equivalent.

6. ## Re: Vertices of equilateral triangle

I second thecmd999's post. For example, let

$\displaystyle b + 37i = re^{i \theta}$

$\displaystyle a + 11i = re^{i \theta + \frac{\pi}{3}$

Dividing the second equation by the first,

$\displaystyle \frac{a+11i}{b+37i} = e^{i \frac{\pi}{3}} = \frac{1 + \sqrt{3}i}{2}$

$\displaystyle (b+37i)(1 + \sqrt{3}i) = 2(a+11i)$

Expand both sides, equate real and imaginary parts. I think I recognize this problem...was it on a past AIME?

7. ## Re: Vertices of equilateral triangle

Yeah, this was 1994 #8.

8. ## Re: Vertices of equilateral triangle

Ah okay. I did AIME in HS (qualified to USAMO once). So most of the past AIME questions I've seen before.