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 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. (OD)/(AD)+(OE)/(BE)+(O)/(CF)= (A) 3/8 (B) 1 (C) 3/2 (D) none of these

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 ABC. The bisector of /_AOB,/_BOC and /_COA meet the sides AB,BC and CA in point D, and F respectively.Show that ADxBExCF =DBxECFA

A triangle ABC is inscribed in a circle with centre at O, The lines AO,BO and CO meet the opposite sides at D,E, and F respectively.Prove that (1)/(AD)+(1)/(BE)+(1)/(CF)=(a cos A+b cos B+cos C)/(phi)

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

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively . Prove that BD = DC.