Question

prove mid point theoram

Answer 1

Extend  to F so that  and join Prove  is a parallelogramIn  and :(1)But these are alternate interior angles, therefore (2)Therefore .We conclude that the line joining the two mid-points of two sides of a triangle is parallel to the third side.Use properties of parallelogram  to prove that (3)We conclude that the line joining the mid-point of two sides of a triangle is equal to half the length of the third side.

Answer 2

Eg with question 
   
                        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)