The problem asks us to prove something about triangle $ABC$ given the equation $\vec{TA} + \vec{TB} + \vec{TC} = \vec{0}$.

1. Problem Description

The problem asks us to prove something about triangle $ABC$ given the equation $\vec{TA} + \vec{TB} + \vec{TC} = \vec{0}$.

2. Solution Steps

Let $O$ be the origin. Then we have:

$\vec{TA} = \vec{OA} - \vec{OT}$

$\vec{TB} = \vec{OB} - \vec{OT}$

$\vec{TC} = \vec{OC} - \vec{OT}$

Adding these equations together, we have:

$\vec{TA} + \vec{TB} + \vec{TC} = \vec{OA} - \vec{OT} + \vec{OB} - \vec{OT} + \vec{OC} - \vec{OT}$

$\vec{TA} + \vec{TB} + \vec{TC} = \vec{OA} + \vec{OB} + \vec{OC} - 3\vec{OT}$

Since $\vec{TA} + \vec{TB} + \vec{TC} = \vec{0}$, we have:

$\vec{0} = \vec{OA} + \vec{OB} + \vec{OC} - 3\vec{OT}$

$3\vec{OT} = \vec{OA} + \vec{OB} + \vec{OC}$

$\vec{OT} = \frac{\vec{OA} + \vec{OB} + \vec{OC}}{3}$

This means that $T$ is the centroid of triangle $ABC$. The centroid of a triangle is the intersection of the medians of the triangle. Thus, the point $T$ is the centroid of the triangle $ABC$.

3. Final Answer

The point $T$ is the centroid of triangle $ABC$.