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The three vertices will be refered to as points A, B and C. A will exist on the innermost circle, B on the middle circle and C on the outer circle.
Since rotation and reflection are excluded, placement of A is arbitrary.
In order to maximize volume, B must be placed exactly opposite A on the middle circle. Earlier proofs conclude that volume to circumference is maximized as the shape approaches a circle, or as the number of vertices approaches infinity. (Can I use other proofs, or do you want this from scratch?)
Connect A, the center of concentricity, and B with a straight line. Draw a line at a right angle at the center. Mark the intersection with the outer circle C. This triangle is unique, and maximizes volume.
Any movements of A or B around their respective circles will narrow the triangle and reduce volume. Any movement of C will skew the triangle and reduce volume.
QED? Or should I call C^2 = A^2 + B^2 - 2AB*cos(c) as my next witness?
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