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` .

D and E are points on equal sides A B and A C of an isosceles triangle A B C such that A D=A E . Prove that B ,C ,D ,E are concylic.

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

If D ,E and F are three points on the sides B C ,CA and A B , respectively, of a triangle A B C show that the (B D)/(C D)=(C E)/(A E)=(A F)/(B F)=-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

If D and E are points on sides A B and A C respectively of a triangle A B C such that D E || B C and B D=C E . Prove that A B C is isosceles.

Da n dE are respectively the points on the side A Ba n dA C of a A B C such that A B=5. 6 c m ,A D=1. 4 c m ,A C=7. 2 c m and A E=1. 8 c m , show that D E ||B Cdot

D , E ,a n dF are the middle points of the sides of the triangle A B C , then (a) centroid of the triangle D E F is the same as that of A B C (b) orthocenter of the tirangle D E F is the circumcentre of A B C

A B C is an isosceles triangle in which A B=A C,B E\ a n d\ C F are its two medians. Show that B E=C Fdot