Prove the quadrilateral property in Exercise 83, assuming the coordinates of P, Q, R, and S are P(x 1 , y 1 , 0), Q(x 2 , y 2 , 0), R(x 3 , y 3 , 0), and S(x 4 , y 4 , z 4 ), where we assume that P, Q, and R lie in the xy plane without loss of generality Data from Exercise 83 The points P, Q, R, an...

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Prove the quadrilateral property in Exercise 83, assuming the coordinates of P, Q, R, and S are

Question:

Prove the quadrilateral property in Exercise 83, assuming the coordinates of P, Q, R, and S are P(x₁, y₁, 0), Q(x₂, y₂, 0), R(x₃, y₃, 0), and S(x₄, y₄, z₄), where we assume that P, Q, and R lie in the xy-plane without loss of generality.

Data from Exercise 83

The points P, Q, R, and S, joined by the vectors u, v, w, and x, are the vertices of a quadrilateral in R³. The four points needn’t lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system.

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