Prove that the four triangles formed by joining in pairs the mid-points of the sides of a triangle are congruent to each other.

Consider the triangle ABC with the points D, E, F as the midpoints of the sides AB, BC and CA respectively.

ED, EF and DF are the midsegments of the triangle so they are each half of 

AC, AB and BC respectively.

thus, 

i)     |AD|=|DB|=|FE|
ii)     |BE|=|CE|=|FD|
iii)     |CF|=|FA|=|ED|

By Side Side Side postulate, the following 4 triangles are congruent:

AFD, FCE, DEB and DEF