Parallelograms and the centroid

A quadrilateral with both pairs of opposite sides equal is a parallelogram. Take the four points A(1, 2, 3) and B(−1, −2, −1), then C(2, 3, 2) with D(4, 7, 6). AB = CD = 6 and BC = DA = √43, so ABCD is a parallelogram.

A parallelogram is a rectangle only if its diagonals are equal. Here AC = √3 and BD = √155, so ABCD is not a rectangle.

Another test: the diagonals of a parallelogram bisect each other, so the midpoints of AC and BD coincide. Both midpoints here are (3/2, 5/2, 5/2).

The centroid of a triangle is the average of its vertices: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3, (z₁ + z₂ + z₃)/3).

Worked: the centroid is (1, 1, 1), A = (3, −5, 7) and B = (−1, 7, −6). Each coordinate of A + B + C must total 3, so C = (3 − 2, 3 − 2, 3 − 1) = (1, 1, 2).

Three dimensional coordinate geometry was developed systematically in the eighteenth century, with Euler taking it up in 1748 after Descartes and Fermat had mainly worked in the plane.