Pompeiu's theorem

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 "B" maps to "B" '. Then we have $scriptstyle\; PC\; =\; P\text{'}C$, and $scriptstyleangle\; PCP\text{'}\; =\; 60^\{circ\}$. Hence triangle "PCP" ' is equilateral and $scriptstyle\; PP\text{'}\; =\; PC$. It is obvious that $scriptstyle\; PA\; =\; P\text{'}B$. 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.

External links

* [http://mathworld.wolfram.com/PompeiusTheorem.html MathWorld's page on Pompeiu's Theorem]