If `D , E ,a n dF` are three points on the sides `B C ,A C ,a n dA B` of a triangle `A B C` such that `A D ,B E ,a n dC F` are concurrent, then show that `BD*CE*AF=DC*EA*FB` .

If D,E and F are three points on the sides BC,CA and AB, respectively,of a triangle ABC such that the (BD)/(CD),(CE)/(AE),(AF)/(BF)=-1

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 a n d E are, respectively, the points on equal sides A B a n d A C of an isosceles triangle A B C such that B , C , E , a n d D are concyclic as shown in Figure. If O is the point of intersection of C D a n d B E , prove that A O is the bisector of line segment D E

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

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.

p A N D q are any two points lying on the sides D C a n d A D respectively of a parallelogram A B C Ddot Show that a r( A P B)=a r ( B Q C)dot

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

Any point D is taken in the base B C of a triangle A B C\ a n d\ A D is produced to E , making D E equal to A Ddot Show that a r\ ( B C E)=a r( A B C)

In an equilateral triangle, prove that the centroid and centre of the circum-circle (circum centre) coincide) Given : An equilateral triangle A B C in which D , E a n d F are the mid-points of sides B C , C A a n d A B respectively. To Prove: The centroid and circum centre are coincident. Construction : Draw medians A D , B E a n d C F

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