try using similarity of triangles..
Just assume the intersection point of diagonal's divide them in the ratio λ:1 and find the position vector of that point using section formula.
Then assume a point on the line joining mid point dividing the line in the ratio μ:1 and find it's position vector. Now, make a case when μ= λ then you'll get the same position vector as in the case of diagonals.
That'll prove that they are concurrent.
(This is a bit long coz I'm not that good at vector's)