Three points, A, B, and C, are chosen at random such that OABC forms a quadrilateral. The midpoints of each edge, P, Q, R, and S, are joined.
Prove that the quadrilateral PQRS will always be a parallelogram.
We shall prove this result by consideration of vectors.
Let= 2a, = 2b, and = 2c.
So= + = -2a + 2b and = + = -2c + 2b.
The position vectors of P, Q, R, and S respectively will be:
= (1/2) = c
= + (1/2) = 2a + -a + b = a + b
= + (1/2) = 2c + -c + b = b + c
Using these we can obtain vectors for each side of the quadrilateral PQRS:
= + = -c + b + c = b
= + = -a + c
= + = -(a + b) + b + c = -a + c
Asis parallel with and equal to , similarly with and , we prove that the quadrilateral PQRS is a parallelogram.
Was it necessary to considerand to prove the result?