Roy's safety-first criterion: Difference between revisions
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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: | |||
:''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.'' | |||
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. | |||
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. | |||
==External links== | |||
*[http://mathworld.wolfram.com/PompeiusTheorem.html MathWorld's page on Pompeiu's Theorem] | |||
[[Category:Elementary geometry]] | |||
[[Category:Triangle geometry]] | |||
[[Category:Articles containing proofs]] | |||
[[Category:Theorems in plane geometry]] | |||
Revision as of 04:23, 24 July 2013
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:
- 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.
The proof is quick. Consider a rotation of 60° about the point C. Assume A maps to B, and P maps to P '. Then we have , and . Hence triangle PCP ' is equilateral and . It is obvious that . Thus, triangle PBP ' has sides equal to PA, PB, and PC and the proof by construction is complete.
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.