D, E, F are mid points of sides BC, CA, AB of `Delta ABC`. Find the ratio of areas of `Delta DEF and Delta ABC`.

ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of Delta ABE and Delta ACD

D, E and F are the mid-points of AB, BC and CA of the equilateral DeltaABC. Prove that DeltaDEF is also an equilateral triangle.

G is the centroid of triangle ABC and A_1 and B_1 are the midpoints of sides AB and AC , respectively. If Delta_1 is the area of quadrilateral G A_1A B_1 and Delta is the area of triangle ABC , then Delta/Delta_1 is equal to a. 3/2 b. 3 c. 1/3 d. none of these

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

If D is the mid point of side BC of a triangle ABCand AD is perpendicular to BC, then

Let D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) of the triangle ABC. Prove that vec(AD)+vec(BE)+vec(CF)=vec(0) .

If L,M,N divide the sides BC,CA and AB of a triangle ABC in the same ratio, then show that the traiangle ABC and dLMN have the same centroid.

B is a vertex of the isosceles triangle ABC. D and E are the mid-points of AB and AC. If BE and CD intersects each other at F prove that DeltaBDE=3DeltaDEF .

D,E,F are the mid - points of the sides overline(BC),overline(CA)andoverline(AB) respectively of the triangle ABC , using coordinate geometry show that , 3(BC^(2)+CA^(2)+AB^(2))=4(AD^(2)+BE^(2)+CF^(2)) .