On the proof of Ptolemy's lemma (Q2012011)

Ptolemy's lemma, stating that, in a cyclic quadrilateral with sides, in order, \(a,b,c,d\) and diagonals \(e\) and \(f\), one has (*) \(ac+bd=ef\), is proved by means of parallelogram area considerations. Given that (*) can be read as a statement regarding the sum of areas, the current proof can be considered to be more explanatory than Ptolemy's own, based on relations derived from some triangle similarity.