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`

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

A B C D is a square E ,\ F ,\ G\ a n d\ H are points on A B ,\ B C ,\ C D\ a n d\ D A respectively, such that A E=B F=C G=D Hdot Prove that E F G H is square.

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

In a A B C ,\ E\ a n d\ F are the mid-points of A C\ a n d\ A B respectively. The altitude A P\ to\ B C intersects F E\ at Qdot Prove that A Q=Q P

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)=2a r(\ F C B)

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)

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)

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\ (B Y X D)=2\ a r\ (\ M B C)

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\ (B Y X D)=a r\ (A B M N)

Let A B C be an isosceles triangle with A B=A C and let D ,\ E ,\ F be the mid points of B C ,\ C A\ a n d\ A B respectively. Show that A D_|_F E\ a n d\ A D is bisected by F E