In ABCD parallelogram M is an arbitrary point. Prove that MA+MC=MB+MD. (MA, MB, MC, MD are vectors)

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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.

 

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