Is a infinite curve?

There is a semicircle with radius $R$ its vertices are called $A$ and $B$ , inside it there are infinite right triangles such that the right angle always lies on the semicircle and hypotenuse is equal to the diameter.

Within each right triangle its inscribed circle is drawn.

If we draw a curve that begins at vertex $A$ and joins to the next center of all inscribed circles , such the curve ends at vertex $B$ . The length $L$ of the curve can be expressed as:

$L = \frac{\sqrt{m}}{n}\times\pi$

$m$ and $n$ are prime numbers

What is $2\times\ m\times\ n$ ?