Ptolemy lived in Alexandria around the year 0. We owe him the next theorem of plane geometry: -
In a cyclic quadrilateral the product of the diagonals equals the sum
of the products of the opposite sides. -
AC.BD = BC.AD + AB.CD
This proof uses similarity of triangles, the Thales lemma and Stewart's theorem. Brief refresher SimilarityTriangles with equal angles are similar, their sides have equal ratio's: Thales lemmaMarked angles are equal bacause they span the same arc of the circle. Stewart's theoremThe proof is [HERE] The theorem of Ptolemy Proof that: -
p.q = a.b + c.d
AE=x and we calculate lines BE and DE and next the theorem of Stewart is applied in ΔABD. Note: stelling = theorem Application Below is pictured a regular heptagon with an intriguing relation: Note: bewijs=proof A very simple proof, thanks to Ptolemy's theorem. This problem was found in FaceBook group "Classical Mathematics". Solution by Kenny Lao. |
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