A triangle `A B C` is inscribed in a circle with centre at `O ,` The lines `A O ,B Oa n dC O` meet the opposite sides at `D , E ,a n dF ,` respectively. Prove that `1/(A D)+1/(B E)+1/(C F)=(acosA+bcosB+ccosC)/`

The bisectors of the angles B and C of a triangle ABC, meet the opposite sides in D and E respectively.If DE|BC, prove that the triangle is isosceles.

An equilateral triangle ABC is inscribed in a circle with centre O. The measures of /_BOC is 30^(@)(b)60^(@)(c)90^(@)(d)120^(@)

In a triangle ABC, the bisector of angle A meets the opposite side at D.Using vectors prove that BD:DC=c:b.

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

The altitudes from the vertices A,B,C of an acute angled triangle ABC to the opposite sides meet the circumcircle at D,E,F respectively . Then (EF)/(BC) =

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

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

In a \ A B C , If L\ a n d\ M are points on A B\ a n d\ A C respectively such that L M B Cdot Prove that: a r\ (\ L O B)=a r\ (\ M O C)

In Figure, A B C D ,\ A B F E\ a n d\ C D E F are parallelograms. Prove that a r( A D E)=a r( B C F)

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90o-1/2A ,90o-1/2B and 90o-1/2C