ABCD is a square. E, F, G and H are the mid points of `AB, BC, CD and DA` respectively. Such that `AE = BF = CG = DH`. Prove that EFGH is a square.

ABCD is quadrilateral E, F, G and H are the midpoints of AB, BC, CD and DA respectively. Prove that EFGH is a parallelogram.

If E, F G and H are respectively the midpoints of the sides AB, BC, CD and AD of a parallelogram ABCD, show that ar(EFGH) =1/2 ar (ABCD) .

if D,E and F are the mid-points of the sides BC,CA and AB respectively of the DeltaABC and O be any points, then prove that OA+OB+OC=OD+OE+OF

ABCD is a rectangle formed by the points A(-1, -1) , B(-1, 4), C(5,4) and D(5, -1) . P,Q R S are the midpoints of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS a square ? a rectangle ? or a rhombus ? Justify your answer.

ABCD is a trapezium with AB||DC . E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB (see the figure). Show that (AE)/(ED)=(BF)/(FC)

ABCD is a convex quadrilateral and 3, 4, 5, and 6 points are marked on the sides AB, BC, CD, and DA, respectively. The number of triangles with vertices on different sides is

If D, E and F are respectively, the mid-points of AB, AC and BC in DeltaABC , then BE + AF is equal to

E and F are respectively the mid-points of equal sides AB and AC of DeltaABC (see figure) Show that BF = CE.

In the figure D, E are points on the sides AB and AC respectively of DeltaABC such that ar(DeltaDBC) = ar(DeltaEBC) . Prove that DE || BC.

In triangle ABC, D is the midpoint of BC. DFZAB and DE bot AC, where points F and E lie on AB and AC respectively. If DF = DE, prove that A ABC is an isosceles triangle.