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.

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) .

Let ABCD is a square with sides of unit length. Points E and F are taken om sides AB and AD respectively so that AE= AF. Let P be a point inside the square ABCD.The maximum possible area of quadrilateral CDFE is-

PQRS is a rectangle formed by joining the points P(-1, -1), Q(-1, 4), R(5,4) and S(5, -1) . A, B, C and D are the mid points of PQ, QR, RS and SR respectively. Is the quadrilateral ABCD a square, a rectangle or a rhombus? Justify your answer.

ABC is right - angled triangle at B. Let D and E be any two point on AB and BC respectively . Prove that AE^2 + CD^2 = AC^2 + DE^2

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

Let ABCD be a square with sides of unit lenght. Points E and F are taken on sides AB and AD, respectively,so that AE=AF . Let P be a point inside the squre ABCD. The value of (PA)^2-(PB)^2+(PC)^2-(PD)^2 is equal to