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Minimal area of a bicentric quadrilateral

A quadrilateral with an incircle is called a tangential quadrilateral and a quadrilateral with a circumcircle is called a cyclic quadrilateral. A bicentric quadrilateral is a convex quadrilateral that is both tangential and cyclic. In [1] we gave a short proof via an inequality that the bicentric quadrilateral with a given incircle and circumcircle that has the maximal area is a right kite. A year after the publication of that note, I received an e-mail from the German mathematician Heinz Schumann asking if I also had a simple proof of the minimal area case. At that time I had not considered that interesting question. Now I have and in this note such a proof is presented. Longer or more complicated proofs can be found in [2, pp. 6-8], [3, pp. 20-25], and [4, pp. 168-171], which uses derivatives.