The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral ABCD, then
where equality holds if and only if the quadrilateral is cyclic. This special case is equivalent to Ptolemy's theorem.
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“A man willing to work, and unable to find work, is perhaps the saddest sight that fortune’s inequality exhibits under this sun.”
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