A very nice ruler and compass geometry puzzle

I found a nice puzzle on Twitter that I want to add some thoughts about here. The problem is that you need to find the shortest route from A to B by touching the perimetre of the circle on the way. We can solve this analytically, by considering a parametric form of the circle and minimizing the distance from the two points but we would ideally require a ruler and compass construction of the solution. Apparently, this problem can be traced way back to Ptolemy in 150 AD.

References:

1. Alhazen’s Billiard Problem on Wolfram Mathworld
2. Circular Billiard by Drexler and Gardner
3. Twitter thread

Figure 1. An example construction of the problem. The goal here is to find the shortest path from A to B while hitting the circumference along the way.

There are three special cases that I want to note here:

1. If B=A, then the shortest route is the round trip from A to the nearest point on the circle.
2. If B or A is on the perimeter, then the shortest route is the straight line from A to B.
3. If B or A is on the center of the circle, then the shortest route is along the radius of the circle.