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| '''Pompeiu's theorem''' is a result of [[plane geometry]], discovered by the Romanian mathematician [[Dimitrie Pompeiu]]. The theorem is quite simple, but not classical. It states the following:
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| :''Given an [[equilateral triangle]] ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a (maybe, degenerate) triangle.''
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| The proof is quick. Consider a rotation of 60° about the point ''C''. Assume ''A'' maps to ''B'', and ''P'' maps to ''P'' <nowiki>'</nowiki>. Then we have <math>\scriptstyle PC\ =\ P'C</math>, and <math>\scriptstyle\angle PCP'\ =\ 60^{\circ}</math>. Hence triangle ''PCP'' <nowiki>'</nowiki> is equilateral and <math>\scriptstyle PP'\ =\ PC</math>. It is obvious that <math>\scriptstyle PA\ =\ P'B</math>. Thus, triangle ''PBP'' <nowiki>'</nowiki> has sides equal to ''PA'', ''PB'', and ''PC'' and the [[proof by construction]] is complete.
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| Further investigations reveal that if ''P'' is not in the interior of the triangle, but rather on the [[circumcircle]], then ''PA'', ''PB'', ''PC'' form a degenerate triangle, with the largest being equal to the sum of the others.
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| ==External links==
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| *[http://mathworld.wolfram.com/PompeiusTheorem.html MathWorld's page on Pompeiu's Theorem]
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| [[Category:Elementary geometry]]
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| [[Category:Triangle geometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in plane geometry]]
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Latest revision as of 15:50, 2 December 2014
Hi there, I am Alyson Boon even though it is not the name on my beginning certification. Distributing production is exactly where her main earnings comes from. I am really fond of handwriting but I can't make it my occupation really. For years she's been living in Kentucky but her spouse wants them to transfer.
My weblog; free tarot readings