I have to solve this problem:
Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0
Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.
I've attached an image which show you what to do in 4 steps. The steps are numbered.
1. Construct the midpoint of AD, call it P.
2. Construct the midpoint of BC, call it Q.
3. Construct the midpoint of PQ: That's M.
If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.
In step #4 you see the complete result.
Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0
if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.
A(a1 , a2)
B(b1 , b2)
C(c1 , c2)
D(d1 , d2)
M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))