How to rotate a vertex around a certain point?

The easiest approach is to compose three transformations:

1. A translation that brings point 1 to the origin
2. Rotation around the origin by the required angle
3. A translation that brings point 1 back to its original position

When you work this all out, you end up with the following transformation (where x is the desired angle of rotation in radians):

newX = centerX + (point2x-centerX)*Math.cos(x) - (point2y-centerY)*Math.sin(x);

newY = centerY + (point2x-centerX)*Math.sin(x) + (point2y-centerY)*Math.cos(x);

Note that this makes the assumption that the angle x is negative for clockwise rotation (the so-called standard or right-hand orientation for the coordinate system). If that's not the case, then you would need to reverse the sign on the terms involving sin(x).

You need a 2-d rotation matrix http://en.wikipedia.org/wiki/Rotation_matrix

Your new point will be

 newX = centerX + ( cosX * (point2X-centerX) + sinX * (point2Y -centerY))
 newY = centerY + ( -sinX * (point2X-centerX) + cosX * (point2Y -centerY))

because you are rotating clockwise rather than anticlockwise

Assuming you are usign the Java Graphics2D API, try this code -

    Point2D result = new Point2D.Double();
    AffineTransform rotation = new AffineTransform();
    double angleInRadians = (angle * Math.PI / 180);
    rotation.rotate(angleInRadians, pivot.getX(), pivot.getY());
    rotation.transform(point, result);
    return result;

where pivot is the point you are rotating around.

1. Translate "1" to 0,0

2. Rotate

   x = sin(angle) * r; y = cos(angle) * r;

3. Translate it back