How do we find the centre of rotation given the original shape and the rotated one?

There are lots of ways, but here's one:

As you rotate the object around some arbitrary center, each point on the object traces out a circular arc. If you were to draw a line segment connecting a point on the shape in its original position to the center of rotation, as well as a line segment connecting the same point after rotation to the center of rotation, along with the arc traced out by the point on the shape, you'd get a "piece of pie" shape as shown with the thin black lines in the figure below.

Obviously you don't know the center of rotation ahead of time, so you wouldn't be able to draw out the slice of pie. That was just for illustrative purposes to set up the next steps. Notice in the following steps that no a priori knowledge of the center of rotation is required.

Replacing the circular arc portion of the pie slice with a simple line segment connecting the point's original location to its new one (shown with the thick black lines in the figure), you'll find that the result is an isosceles triangle.

Now we can draw a line perpendicular to the base of the triangle passing through the base's midpoint (shown with the green lines). Since the triangle is isosceles, that line must pass through the vertex opposite the base, which happens to be the center of rotation.

If you draw two such lines, corresponding to two different points on your shape, their intersection must be the center of rotation.

So here's the procedure:

1) Find the before and after coordinates for a point on the shape.

2) Find the midpoint of the before and after coordinates, as well as the slope of the line connecting them.

3) Find the equation of a perpendicular (i.e. negative reciprocal of the slope) line passing through that midpoint.

4) Repeat 1-3 for a second point on the shape.

5) Set the two linear equations equal to each other to find the point of intersection.

Thanks for the clear explanation, it really helps