Largest Triangle inside a Curvilinear Triangle

Samuel Prime

RegionUnderCircleIn the diagram shown we have part of a circle of radius $latex R$ whose center is at the point $latex (R,R)$ 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…

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