check if point is on a plane (using Heron formula ?)

Question

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane?

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I am given 4 points in 3 dimensional space. Is this correct to state that all 4 points belong to some, one and the same plane if and only if a content

of a tetrahedron with vertices being these points is 0 (zero) ?

Heron-type formula for the volume of a tetrahedron is

where U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; u opposite to U and so on). This is derived from the Cayley–Menger determinants.

So one can test therefore if Volume = 0, thus if any of parameters a, b, c, d is equal to sum of three others?

Answer 1

When the lengths of the edges of the tetrahedron are given, you can indeed calculate the volume of the tetrahedron (assuming the formula you gave is correct) and check whether or not is equals $0$. It may be easier however to just calculate the areas of the four faces of the tetrahedron (using Herons formula) and check whether one area equals the sum of the other three.

Answer 2

Four points, $(x_0\mid y_0\mid z_0)$, $(x_1\mid y_1\mid z_1)$, $(x_2\mid y_2\mid z_2)$ and $(x_3\mid y_3\mid z_3)$, are coplanar if the determinant $\begin{vmatrix} x_0&y_0&z_0&1\\ x_1&y_1&z_1&1\\ x_2&y_2&z_2&1\\ x_3&y_3&z_3&1\\ \end{vmatrix}$ vanishes.