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

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 a rectangle and P,R and S are the mid - points of the AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that vec(AB)+vec(AD)+vec(CB)+vec(CD)=4vec(EF) .

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 (A) 270 (B) 220 (C) 282 (D) 342

In DeltaABC, D, E and F are the midpoints of sides AB, BC and CA respectively. Show that DeltaABC is divided into four congruent triangles, when the three midpoints are joined to each other. (ΔDEF is called medial triangle)

ABCD a parallelogram, and A_1 and B_1 are the midpoints of sides BC and CD, respectively. If vec(A A)_1 + vec(AB)_1 = lamda vec(AC) , then lamda is equal to

ABCD is a rectangle with A as the origin. vec b and vec d are the position vectors of B and D respectively. (i) What is the position vector of C? (ii) If P, Q, R and S are midpoints of sides of AB, BC, CD and DA respectively, find the position vector of P, Q, R,S,

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

ABCD is a quadrilateral in which AB= AD, the bisector of angle BAC and angle CAD intersect the sides BC and CD at the points E and F respectively. Prove that EF || BD.