If `D ,Ea n dF` are three points on the sides `B C ,C Aa n dA B ,` respectively, of a triangle `A B C` such that the `(B D)/(C D),(C E)/(A E),(A F)/(B F)=-1`

In triangle A B C ,poin t sD , Ea n dF are taken on the sides B C ,C Aa n dA B , respectigvely, such that (B D)/(D C)=(C E)/(E A)=(A F)/(F B)=ndot Prove that _(D E F)=(n^2-n+1)/((n+1)^2)_(A B C)dot

If D ,E ,F are the mid-points of the sides B C ,C Aa n dA B respectively of a triangle A B C , prove by vector method that A r e aof D E F=1/4(a r e aof A B C)dot

D ,\ E ,\ F are the mid-points of the sides B C ,\ C A and A B respectively of a A B C . Determine the ratio of the areas of D E F and A B C .

If O is a point in space, A B C is a triangle and D , E , F are the mid-points of the sides B C ,C A and A B respectively of the triangle, prove that vec O A + vec O B+ vec O C= vec O D+ vec O E+ vec O Fdot

D ,\ E ,\ F are the mid-point of the sides B C ,\ C A\ a n d\ A B respectively of A B Cdot Then D E F is congruent to triangle. A B C (b) AEF (c) B F D ,\ C D E (d) A F E ,\ B F D ,\ C D E

D ,\ E and F are the points on sides B C ,\ C A and A B respectively of A B C such that A D bisects /_A ,\ \ B E bisects /_B and C F bisects /_C . If A B=5c m ,\ \ B C=8c m and C A=4c m , determine A F ,\ C E and B D .

In a A B C ,\ D ,\ E ,\ F are the mi-points of sides B C ,\ C A\ a n d\ A B respectively. If a r( A B C)=16 c m^2, then a r\ (t r a p e z i u m\ F B C E)= 4\ c m^2 (b) 8\ c m^2 (c) 12 c m^2 (d) 10 c m^2

Let A B C be an isosceles triangle in which A B=A Cdot If D ,\ E ,\ F be the mid-points of the sides B C ,\ C A\ a n d\ A B respectively, show that the segemtn A D\ a n d\ E F bisect each other at right angles.

In a parallelogram A B C D ,E ,F are any two points on the sides A B and B C respectively. Show that a r( A D F)=a r( D C E)dot

In Figure, A B C is a right triangle right angled at A ,\ B C E D ,\ A C F G\ a n d\ A M N are square on the sides B C ,\ C A\ a n d\ A B respectively. Line segment A X\ _|_D E meets B C at Ydot Show that: a r(C Y X E)=a r\ (A C F G)