1. **Problem statement:** (i) Show that the centroid of the face opposite to vertex $t$ in a tetrahedron with vertices $t, u, v, w$ is $\frac{1}{3}(u+v+w)$, and write down the centroids of the other three faces. (ii) Show that the point $\frac{3}{4}$ of the way from $t$ to the centroid of the opposite face is $\frac{1}{4}(t+u+v+w)$, the centroid of the tetrahedron. (iii) Deduce that the four lines from each vertex to the centroid of the opposite face meet at the centroid of the tetrahedron. --- 2. **Formula and rules:** - The centroid of a triangle with vertices $A, B, C$ is given by $\frac{1}{3}(A + B + C)$. - The centroid of a tetrahedron with vertices $t, u, v, w$ is $\frac{1}{4}(t + u + v + w)$. - A point $p$ fraction $\lambda$ along the line from $A$ to $B$ is $p = (1-\lambda)A + \lambda B$. --- 3. **Step (i): Centroids of faces** - The face opposite to $t$ has vertices $u, v, w$. - Its centroid is: $$\frac{1}{3}(u + v + w)$$ - Similarly, the face opposite to $u$ has vertices $t, v, w$ with centroid: $$\frac{1}{3}(t + v + w)$$ - The face opposite to $v$ has vertices $t, u, w$ with centroid: $$\frac{1}{3}(t + u + w)$$ - The face opposite to $w$ has vertices $t, u, v$ with centroid: $$\frac{1}{3}(t + u + v)$$ --- 4. **Step (ii): Point $\frac{3}{4}$ from $t$ to centroid of opposite face** - The centroid of the face opposite $t$ is $C = \frac{1}{3}(u + v + w)$. - The point $P$ that is $\frac{3}{4}$ of the way from $t$ to $C$ is: $$P = (1 - \frac{3}{4})t + \frac{3}{4}C = \frac{1}{4}t + \frac{3}{4} \times \frac{1}{3}(u + v + w)$$ - Simplify: $$P = \frac{1}{4}t + \frac{1}{4}(u + v + w) = \frac{1}{4}(t + u + v + w)$$ - This is exactly the centroid of the tetrahedron. --- 5. **Step (iii): Deduction about concurrency** - By symmetry, the same applies for the lines from $u, v, w$ to the centroids of their opposite faces. - Each such point $\frac{3}{4}$ along the line from a vertex to the opposite face centroid equals the tetrahedron centroid $\frac{1}{4}(t + u + v + w)$. - Therefore, the four lines from each vertex to the centroid of the opposite face intersect at the centroid of the tetrahedron. --- **Final answers:** - Centroids of faces: - Opposite $t$: $\frac{1}{3}(u + v + w)$ - Opposite $u$: $\frac{1}{3}(t + v + w)$ - Opposite $v$: $\frac{1}{3}(t + u + w)$ - Opposite $w$: $\frac{1}{3}(t + u + v)$ - The point $\frac{3}{4}$ along the line from $t$ to its opposite face centroid is the tetrahedron centroid: $$\frac{1}{4}(t + u + v + w)$$ - The four lines from vertices to opposite face centroids concur at the tetrahedron centroid.