The midpoint of two opposite sides of a quadrilateral and the midpoint of the diagonals are the vertices of a parallelogram. Prove that using vectors.

Explanation

Let `veca, vecb, vecc and vecd` be the position vectors of vertices A, B, C and D, respectively.
Let E, F, G and H be the midpoints of AB, CD, AC and BD, respectively.
`" "P.V. "of "E=(veca+vecb)/(2)`
`" "P.V. "of "F=(vecc+vecd)/(2)`
`" "P.V. "of "G=(veca+vecc)/(2)`
`" "P.V. "of "H = (vecb+vecd)/(2)`
`" " vec(EG)= P.V. " of" G-P.V. "of "E=(veca+vecc)/(2)-(veca+vecb)/(2)=(vecc-vecb)/(2)`
`" "vec(HF)=P.V. "of "F-P.V. "of "H=(vecc+vecd)/(2)-(vecb+vecd)/(2)=(vecc-vecb)/(2)`
`therefore" "vec(EG)=vec(HF)rArrEG"||"HF and EG=HF`
Hence, EGHF is a parallelogram.