Chapter 8.2, Problem 29E

In Exercises 21-24, a, b, and c are noncollinear points in ℝ² and p is any other point in ℝ². Let Δabc denote the closed triangular region determined by a, b, and c, and let Δpbc be the region determined by p, b, and c. For convenience, assume that a, b. and c are arranged so that det $\left[\begin{array}{ccc}\tilde{\text{a}}& \tilde{\text{b}}& \tilde{\text{c}}\end{array}\right]$ is positive, where $\tilde{\text{a}}$, $\tilde{\text{b}}$, and $\tilde{\text{c}}$ are the standard homogeneous forms for the points.

21. Show that the area of Δabc is det $\left[\begin{array}{ccc}\tilde{\text{a}}& \tilde{\text{b}}& \tilde{\text{c}}\end{array}\right]/2$. [Hint: Consult Sections 3.2 and 3.3, including the Exercises.]