A recap of the proof of Ptolemy’s theorem I did in class today. This is a well known proof. Simple and constructive in nature.

Given cyclic quadrilateral ABCD as shown. From point C, make line CE such that ∠ECD ≌ ∠BCA.

One can easily see that

ΔABC ∽ ΔDEC

which implies AB : DE = AC : DC … (1)

Also because ΔCAD ∽ ΔCBE,

which implies BC : AC = BE : AD … (2)

By (1) : AB * DC = AC * DE

By (2) : BC * AD = AC * BE Adding these two equations gives

AB * DC + BC * AD = AC * DE + AC * BE = AC * (DE + BE) = AC * DB Voila.

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## dedusuiu said

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## dudiasu said

in many problems can be observed that theorems of Desargues considered prof dr nircea orasanu and prof drd horia orasanu can be applied to other domains as Constraints Optimizations with non holonomic aspects and prof dr Gigel Militaru with his doctoral students appear and thus these lead to Adrien Legendre theorems and results