Calculate middle point of Bezier Curve

I think this is what you're looking for:

http://blog.sklambert.com/finding-the-control-points-of-a-bezier-curve/

It goes into detail on calculating the various points on a Bezier curve.

You may also be interested in this more specific example for your application:

http://www.codeproject.com/Articles/223159/Midpoint-Algorithm-Divide-and-Conquer-Method-for-D

If you really want to get into it, then I suggest this Primer:

http://pomax.github.io/bezierinfo/

Bezier curves are a bit more complicated than simple arcs. For an arc, you can just use this formula:

R = H/2 + W^2/8H

...which definitely won't work for a Bezier curve. On a Quadratic Bezier curve, for example, to calculate a point, you must use:

Sources: http://en.wikipedia.org/wiki/B%C3%A9zier_curve, Quadratic Bezier Curve: Calculate Point

Below is what I use to get the control point of a quad bezier curve. It should work for your problem where the control point is on the curve. It's in Swift but you should be able to convert it to another language easily. Basically at the midpoint of the line (whose points are point1 and point2) I work out a perpendicular line with the given length. Clockwise parameter determines which side of the line the point should fall on.

func getControlPointWithPoint1(point1:CGPoint, point2:CGPoint, length:CGFloat, clockwise:Bool) -> CGPoint {
  let angle = getAngleWithPoint1(point1, point2:point2)
  let direction = clockwise ? 1 : -1
  let perpendicularAngle = angle + (CGFloat(direction) * CGFloat((M_PI / 2)))
  let midPoint = getMidPointWithPoint1(point1, point2:point2)
  return CGPointMake(midPoint.x + (cos(perpendicularAngle) * length), midPoint.y + (sin(perpendicularAngle) * length))
}

func getAngleWithPoint1(point1:CGPoint, point2:CGPoint) -> CGFloat {
  return atan2((point2.y - point1.y), (point2.x - point1.x))
}

func getMidPointWithPoint1(point1:CGPoint, point2:CGPoint) -> CGPoint {
  return CGPointMake((point1.x + point2.x) / 2, (point1.y + point2.y) / 2)
}

Below is how it would map to your diagram letters:

c = getControlPointWithPoint1(a, point2:b, length:h, clockwise:true)