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prove that the line joining the mid points of two is parallel to the third side and equal to half the length of the third side

Draw a triangle & extend DE to F so that DE=EF and join FC

prove BCFD is a parallelogram

in triangle EAD and triangle ECF

E1 = E2 ( vert. opp.angle s)

AE=CE(given)

DE=EF(by construction)

therefor triangle EAD=triangle ECF(SAS)