Chapter 8.3, Problem 10E

Repeat Exercise 9 for the points q₁, …, q₅ whose barycentric coordinates with respect to S are given by $\left(\frac{1}{8},\frac{1}{4},\frac{1}{8},\frac{1}{2}\right)$, $\left(\frac{3}{4},-\frac{1}{4},0,\frac{1}{2}\right)$, $\left(0,\frac{3}{4},\frac{1}{4},0\right)$, (0, −2, 0, 3), and $\left(\frac{1}{3},\frac{1}{3},\frac{1}{3},0\right)$, respectively.

9. Let S = {v₁, v₂, v₃, v₄} be an affinely independent set. Consider the points p₁, …, p₅ whose barycentric coordinates with respect to S are given by (2, 0, 0, −1), $\left(0,\frac{1}{2},\frac{1}{4},\frac{1}{4}\right)$, $\left(\frac{1}{2},0,\frac{3}{2},-1\right)$, $\left(\frac{1}{3},\frac{1}{4},\frac{1}{4},\frac{1}{6}\right)$, and $\left(\frac{1}{3},0,\frac{2}{3},0\right)$, respectively. Determine whether each of p₁, …, p₅ is inside, outside, or on the surface of conv S, a tetrahedron. Are any of these points on an edge of conv S?