In parallelogram ABCD, M is an arbitrary point. Prove that MA+MC=MB+MD.
(MA, MB, MC, MD are vectors)
Start with MA+MC. Note that we can use the head-to-tail rule to write MA as MD+DA, and MC as MD+DC. Since vector addition is commutative and substituting for MA and MC we have
MA+MC = MD + DA + MD + DC
= MD + MD + DA + DC
Since ABCD is a parallelogram, DA + DC = DB so we now have
= MD + MD + DB
But MD + DB = MB by the head-to-tail rule, thus we have
= MD + MB , and again using the commutative property
= MB + MD as required.