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

If D is the midpoint of the side AB of a triagle ABC prove that vec(BC)+vec(AC)=-2vec(CD)

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

If G is the centroid of a triangle ABC, prove that vec(GA)+vec(GB)+vec(GC)=vec(0) .

(Apollonius theorem): If D is the midpoint of the side BC of a triangle ABC, then show by vector method that |vec(AB)|^(2) + |vec(AC)|^(2) = 2(|vec(AD)|^(2)+ |vec(BD)|^(2)).

If O( vec0 ) is the circumcentre and O' the orthocentre of a triangle ABC, then prove that i. vec(OA)+vec(OB)+vec(OC)=vec(OO') ii. vec(O'A)+vec(O'B)+vec(O'C)=2vec(O'O) iii. vec(AO')+vec(O'B)+vec(O'C)=2 vec(AO)=vec(AP) where AP is the diameter through A of the circumcircle.

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that vec(AB)+vec(AD)+vec(CB)+vec(CD)=4vec(EF) .

ABCD a parallelogram, and A_1 and B_1 are the midpoints of sides BC and CD, respectively. If vec(A A)_1 + vec(AB)_1 = lamda vec(AC) , then lamda is equal to

D is the midpoint of the side BC of Delta ABC . If P and Q are points o AB and on AC such that DP bisects angle BDA and DQ bisects angle ADC , then prove that PQ || BC