1. **Problem Statement:** Show that the center of a square with vertices $t$, $u$, $v$, and $w$ is given by the point $\frac{1}{4}(t + u + v + w)$, where the center is equidistant from all vertices. 2. **Key Concept:** The center of a square is the point equidistant from all four vertices. This point is also the intersection of the diagonals. 3. **Formula for the center (centroid) of a quadrilateral:** $$\text{Center} = \frac{1}{4}(t + u + v + w)$$ This formula calculates the average of the position vectors of the vertices. 4. **Proof:** - Since the square has vertices $t$, $u$, $v$, and $w$ in order, the diagonals are $t$ to $v$ and $u$ to $w$. - The midpoint of diagonal $t$ to $v$ is: $$M_1 = \frac{t + v}{2}$$ - The midpoint of diagonal $u$ to $w$ is: $$M_2 = \frac{u + w}{2}$$ - In a square, the diagonals bisect each other, so the center is the midpoint of both diagonals, meaning $M_1 = M_2$. 5. **Calculate the average of all vertices:** $$\frac{1}{4}(t + u + v + w) = \frac{1}{2} \left( \frac{t + v}{2} + \frac{u + w}{2} \right) = \frac{1}{2}(M_1 + M_2)$$ Since $M_1 = M_2$, this simplifies to: $$\frac{1}{2}(M_1 + M_1) = M_1$$ 6. **Conclusion:** The point $\frac{1}{4}(t + u + v + w)$ coincides with the midpoint of the diagonals, which is the center of the square, equidistant from all vertices. **Final answer:** $$\boxed{\text{Center} = \frac{1}{4}(t + u + v + w)}$$