O is any point in the plane of the triangle ABC,AO,BO and CO meet the sides BC,CA nd AB in D,E,F respectively show that `(OD)/(AD)+(OE)/(BE)+(OF)/(CF)=1.`

Consider /_\ ABC Let I be the incentre and a,b c be the sides of the triangle opposite to the angle A,B,C respectively. Let O be any point in the plane of /_\ ABC within the triangle . AO ,BO ,CO meet the sides BC, CA and AB in D, E and F respectively then (OD)/(AD) +(OE)/(BE)+ (OF)/(CF) =

Consider /_\ABC . Let I bet he incentre and a,b,c be the sides of the triangle opposite to angles A,B,C respectively. Let O be any point in the plane of /_\ABC within the triangle. AO,BO and CO meet the sides BC, CA and AB in D,E and F respectively. If 3vec(BD)=2vec(DC) and 4vec(CE)=vec(EA) then the ratio in which divides vec(AB) is (A) 3:4 (B) 3:2 (C) 4:1 (D) 6:1

Consider /_\ABC . Let I bet he incentre and a,b,c be the sides of the triangle opposite to angles A,B,C respectively. Let O be any point in the plane of /_\ABC within the triangle. AO,BO and CO meet the sides BC, CA and AB in D,E and F respectively. avec(IA)=bvec(IB)+cvec(IC)= (A) -1 (B) 0 (C) 1 (D) 3

O is any point inside a triangle A B C . The bisector of /_A O B ,/_B O C and /_C O A meet the sides A B ,B C and C A in point D ,E and F respectively. Show that A D * B E * C F=D B * E C * F A

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 any point on the BC of an isosceles triangle ABC then prove that AB gt AD and AC gt AD .

X Y is a line parallel to side B C of triangle A B C. BE ll AC and CF ll AB meet XY in E and F respectively. Show that a r( A B E)=a r( A C F)dot

The incircle of an isoceles triangle ABC, with AB=AC, touches the sides AB,BC and CA at D,E and F respecrively. Prove that E bisects BC.

D is a point in side BC of triangle ABC. If AD gt AC , show that AB gt AC .

In Delta ABC, AB = AC. D , E and F are mid-points of the sides BC, CA and AB respectively . Show that : AD is perpendicular to EF.