Proving Triangle Midpoint Sum Equals Side Length Sum

[vg19]

Hey,

I am struggling with one question in vector proofs.

If D, E, and F are the midpoints of the sides of a triangle ABC, prove that
OD + OE + OF = OA + OB + OC

I don't really understand what we are trying to prove here. My first thought was that from some origin, to the points D E and F added up, is the same distance from some origin to A B and C added up.

I would really apprcieate some help. Also are there any tips or techniques you can give for generally solving questions that ask for proof?

Thanks

 

[Leong]

is the equation in vector notation or it is a normal algebra equation ?

 

[vg19]

vector notation...sorry couldn't draw the arrows, and acutally i do have 1 more qusetion that I ma not getting an answer to,

Prove that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Ive been able to start saying that

AB must be parallel and equal to DC
AD must be parallel and equal to BC

However, I am not sure how to prove this.

Any help is apprcieated

 

[Leong]

let\ A(a_x,a_y), B(b_x,b_y), C(c_x,c_y)
find\ \vec{OA}+\vec{OB}+\vec{OC}in terms of \hat{i} \ and \hat{j}
find the coordinate of D,E and F using the midpoint formula in term of the coordinates of A, B or C and find
find\ \vec{OD}+\vec{OE}+\vec{OF}in terms of \hat{i} \ and \hat{j}

You should get the same result.
I have tried it.