## Task A-3.6.

Point $M$ is the midpoint of side $\overline{A B}$, and $T$ is the centroid of triangle $A B C$. If $A M T$ is an equilateral triangle with side length 1, determine the lengths of the sides of triangle $A B C$.
## First Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_ce71b02a7f068198719dg-20.jpg?height=763&width=716&top_left_y=481&top_left_x=590)

Since $|M B|=|M A|=1$, it follows that $|A B|=2$.

1 point

Furthermore, since $|A M|=|T M|=|B M|$, it follows that $M$ is the circumcenter of triangle $A B T$, and by Thales' theorem, $\varangle A T B=90^{\circ}$. Therefore, by the Pythagorean theorem, $|B T|=\sqrt{|A B|^{2}-|A T|^{2}}=\sqrt{3}$.

Let $K$ and $L$ be the midpoints of sides $\overline{B C}$ and $\overline{C A}$, respectively. Since the centroid divides the median in the ratio 2:1, it follows that $|T L|=\frac{1}{2}|B T|=\frac{\sqrt{3}}{2}$ and $|T K|=\frac{1}{2}|A T|=\frac{1}{2}$.

From the right triangles $B T K$ and $A T L$, we get

$$
\begin{aligned}
& |B K|=\sqrt{|B T|^{2}+|T K|^{2}}=\frac{\sqrt{13}}{2} \\
& |A L|=\sqrt{|A T|^{2}+|T L|^{2}}=\frac{\sqrt{7}}{2}
\end{aligned}
$$

from which it follows that $|B C|=2|B K|=\sqrt{13}$ and $|A C|=2|A L|=\sqrt{7}$.