Mathematics

When four points lie in the same plane, their position vectors satisfy a linear dependence—one point can be expressed as a combination of the others. The key is to write $\mathbf{d}$ in terms of $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, then substitute into a scalar triple product. This transforms the left side into a sum of three products on the right, each representing how $\mathbf{d}$'s "weight" distributes across the triangular regions formed by the other three points.

When four points lie in the same plane, their position vectors satisfy a linear dependence—one point can be expressed as a combination of the others. The key is to write $\mathbf{d}$ in terms of $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$, then substitute into a scalar triple product. This transforms the left side into a sum of three products on the right, each representing how $\mathbf{d}$'s "weight" distributes across the triangular regions formed by the other three points.