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Generalizations of Ptolemy and Brahmagupta Theorems
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Journal Article
Generalizations of Ptolemy and Brahmagupta Theorems
2007
Overview
The Greek astronomer Ptolemy of Alexandria (second century) and the Indian mathematician Brahmagupta (sixth century) each have a significant theorem named after them. Both theorems have to do with cyclic quadrilaterals. Ptolemy's theorem states that: In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of two pairs of opposite sides. If the lengths of the sides of a cyclic quadrilateral are a, b, c, d in this order and the lengths of the diagonals are l and k, then Ptolemy's theorem would be expressed as lk = ac + bd. On the other hand, Brahmagupta came up with a remarkable formula for the area E of the cyclic quadrilateral, that is E = [square root](s-a)(s-b)(s-c)(s-d), where s stands for the semiperimeter 1/2(a+b+c+d). Around the middle of the 19th century, there appeared generalizations for both theorems that apply to any convex quadrilateral. The German mathematicians C. A. Bretschneider and F. Strehlke each published their own proofs of the generalizations. Since then, more proofs have shown up in the literature. This paper presents new proofs, which could be used in the classroom or as projects outside the classroom. In addition, it looks into some implications of the two generalizations, and shows that they are not independent of each other. (Contains 1 figure.)
Publisher
MATYC Journal Inc,Mathematics and Computer Education
Subject
