Three vertices of a triangle are `A (1, 2), B (-3, 6) and C(5, 4)`. If D, E and F are the mid-points of the sides opposite to the vertices A, B and C, respectively, show that the area of triangle ABC is four times the area of triangle DEF.

Thevertices of a triangle ABC are A(-7, 8), B (5, 2) and C(11,0) . If D, E, F are the mid-points of the sides BC, CA and AB respectively, show that DeltaABC=4 DeltaDEF .

If D, E and F are the mid-points of the sides of an equilateral triangle ABC, then the ratio of the area of triangle DEF and DCF is :

ABC is a triangle in which D, E and F are the mid-points of the sides AC, BC and AB respectively. What is the ratio of the area of the shaded to the unshaded region in the triangle?

In the adjoining figure D, E and F are the mid-points of the sides BC, AC and AB respectively. triangle DEF is congruent to triangle :

Consider a triangle ABC and let a,b and c denote the lengths of the sides opposite to vertices A,B, and C, respectively.Suppose a=6,b=10, and the area of triangle is 15sqrt(3.) If /_ACB is obtuse and if r denotes the radius of the incircle of the triangle,then the value of r^(2) is