arithmetic problem concerning equilateral triangle An equilateral triangle exists with vertices $(0,0), (a,11)$ and $(b,37)$. Using the distance formula three times, I eventually arrive to:
$$a^2 + 121 = b^2 + 1369 = a^2 + b^2 - 2ab + 676$$
The only problem is that since none of the equations are actually equal to anything, I can't figure it out. How can I solve this? 

You actually have two equations: for instance, 
\begin{align}
a^2+121&=b^2+1369\\
b^2+1369&=a^2+b^2-2ab+676.
\end{align}
It is easy to see that $b=0$ gives no solution: the second equation becomes $121=676$. So we assume $b\ne0$. We rewrite the equations as 
\begin{align}
a^2-b^2&=1248\\
a^2-2ab&=693.
\end{align}
If you subtract both equations, you get 
$$
-b^2+2ab=555.
$$
Thus,
$$
a=\frac{555+b^2}{2b}.$$ If we plug this into $a^2-b^2=1248$,
$$
1248=\frac{555^2+b^4+1110b^2}{4b^2}-b^2
=\frac{555^2-3b^4+1110b^2}{4b^2},
$$
so
$$
3b^4+3882b^2-555^2=0.
$$
Now
\begin{align}
b^2&=\frac{-3882\pm\sqrt{3882^2+4\times3\times555^2}}{6}
=\frac{-3882\pm\sqrt{2^4\times 3^2\times 19^4}}{6}\\ &=\frac{-3882\pm2^2\times3\times 19^2}{6}
=647\pm722,
\end{align}
so $b^2=75$ (the negative solution is not acceptable because it won't give $b$ real), and then $b=\pm5\sqrt3$. 
Correspondingly, 
$$
a=\pm\sqrt{1248+b^2}=\pm\sqrt{1248+75}=\pm\sqrt{1323}=\pm21\sqrt3.
$$
