If P is a point inside `Delta ABC` such that `BC (vec(PA))+CA(vec(PB))+AB(vec(PC))= vec O,` then P is the ......

Let ABC be a triangle and P be a point inside ABC such that vec(PA) + 2vec(PB) + 3vec(PC) = vec0 . The ratio of the area of triangle ABC to that of APC is - (A) 2 (B) 3/2 (C) 5/3 (D) 3

D, E, F are the midpoints of the sides bar(BC), bar(CA) and bar(AB) respectively of the triangle ABC. If P is any point in the planeof the triangle, show that vec(PA) + vec(PB) +vec(PC) = vec(PD) + vec(PE) +vec(PF).

G is a point inside the plane of the triangle ABC, if vec(GA)+vec(GB)+vec(GC)=vec(0) , then show that G is the centroid of the triangle ABC.

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

In triangle ABC, find vec(AB)+vec(BC)+vec(CA)

If D is the midpoint of the side BC of a triangle ABC, prove that vec(AB)+vec(AC)=2vec(AD)

If O is the circumcentre and P the orthocentre of Delta ABC , prove that vec(OA)+ vec(OB) + vec(OC) =vec(OP) .

If D is the mid-point of the side BC of a triangle ABC, prove that vec AB+vec AC=2vec AD .

If D, E, F are the midpoints of BC, CA, AB of Delta ABC, then vec(A)D + vec(B)E + vec(C)F =