Hi,

You will find a point lying on the curve somewhere near the center by calculating the Bezier formula with the t parameter set to 0.5:

x(t) = (1-t)^3 x0 + 3t(1-t)^2 x1 + 3t^2 (1-t) x2 + t^3 x3
y(t) = (1-t)^3 y0 + 3t(1-t)^2 y1 + 3t^2 (1-t) y2 + t^3 y3

Xi and Yi above are the control points in a Bezier segment.

If you have more than one segment, you can do that calculation for the middle segment when there are odd number of segments, or just take the middle control point when there are even number of segment.

That won't find the ideal middle point, but at least the point will lie on the Bezier curve. Finding the ideal middle point for a Bezier involves some calculus and solving integrals 

Stoyan