TRIANGLES SIMILARITY

the figure formed by joining the mid-points of pairs of consecutive sides of a quadrilateral is a parallelogram.

Answer:    Given:
ABCD is a quadrilateral. E, F, G and H are the mid-points of
AB, BC, CD and DA respectively.
To prove:
EFGH is a parallelogram.
Construction:
Join A to C and D to B.
Proof:
Consider D ACD,

mid-points of two sides of a triangle is parallel to the third side and half of it)
Consider D ACB,

mid-points of two sides of a triangle is parallel to the third side and half of it)
From (i) and (ii),
HG=EF and HG||EF
\ EFGH is a parallelogram.

If AC=BD   then EFGH is a rhombus.