Computation of r and A using geometry and trigonometry

From the isoceles triangle with base s, we get that

Combining this result with the two similar triangles involving the angle A, we have

Next observe that

Now use the Law of Cosines to solve for r:

Finally, the Law of Sines (and some help from Mathematica for the calculation) allows us to determine the angle A:

Whew, how's that for an application of geometry and trigonometry!