Triangle ABC and parallelogram ABEF are on the same base, AB as in between the same parallels AB and EF. Prove that ar `(DeltaABC) = 1/2` ar(|| gm ABEF)

If a triangle and a parallelogram lie on the same base and between the same parallels, then prove that the area of the triangle is equal to half of the area of parallelogram.

Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

ABCD is parallelogram and ABEF is a rectangle and DG is perpendicular on AB. Prove that (i) ar (ABCD) = ar(ABEF) (ii) ar (ABCD) = AB xx DG

ABC is an isosceles triangle right angled at C. Prove that AB^(2) = 2AC^(2) .

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

ABC is a right triangle , right angled at A and D is the mid point of AB . Prove that BC^(2) =CD^2 +3BD^(2) .

The diameters of the circumcirle of triangle ABC drawn from A,B and C meet BC, CA and AB, respectively, in L,M and N. Prove that (1)/(AL) + (1)/(BM) + (1)/(CN) = (2)/(R)

Vertices of a parallelogram taken in order are A, ( 2,-1,4) , B (1,0,-1) , C ( 1,2,3) and D. The distance between the parallel lines AB and CD is

Let A(p^2,-p),B(q^2, q),C(r^2,-r) be the vertices of triangle ABC. A parallelogram AFDE is drawn with D,E, and F on the line segments BC, CA and AB, respectively. Using calculus, show that the maximum area of such a parallelogram is 1/2(p+q)(q+r)(p-r)dot

Two triangles, DeltaABC" and "DeltaDBC, lie on the same side of the base BC. From a point P on BC, PQ||AB" and "PR||BD are drawn. They intersect AC at Q DC at R. Prove that QR||AD.