## Largest Triangle inside a Curvilinear Triangle

In the diagram shown we have part of a circle of radius whose center is at the point and which is tangent to the x and y axes — though the graph is drawn for R = 2, we want to work with general R.

Our focus is on the region under the circle above the x-axis. The question is: what is the maximum area that a triangle inside this region can have?

It may occur to you that there is a reasonable `quick’ answer, but the point of the problem is to reason it out carefully so you more or less have a proof that you do indeed get a maximum area. Since the region is concave, the vertices of a triangle cannot be so that one is too close to the far right while another vertex close to the far top left (or else the triangle would not be fully inside the region).

Have fun!

## Two Little Circles

In the diagram shown we see a big circle of radius R that is tangent to both the x and y axes. What is the radius of the little circle to its southwestern corner? Express its radius in terms of R. (The little circle is also required to be tangent to the axes. The diagram is draw for a circle of radius 2 but we’re working with a general radius R.)

After having solved this problem, the next question will be this:

You can easily see that we can repeat the process by looking at the southwestern circles for ever and ever because there will always be a gap, and you get smaller and smaller circles with small radii. What if you add up all their radii? Do they add up? If so to what number do they add up, in terms of the original radius R. If they don’t add up, why don’t they?