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Pompeiu's Theorem


If P be a point in the plane of an equilateral triangle DeltaABC, then the lengths of line segments AP, BP, and CP correspond the sides of a triangle, which is degenerate when P lies on the circumcircle of DeltaABC.

This theorem is an immediate consequence of the Ptolemy inequality.

Pompeiu's theorem stays valid when P is outside the plane of ABC in the Euclidean three-space (Veldkamp 1956-1957).


See also

Ptolemy Inequality, Tweedie's Theorem

This entry contributed by Floor van Lamoen

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References

Boomstra, W. "Nogmaals de stelling van Pompeïu." Nieuw Tijdschr. Wisk. 44, 285-288, 1956-1957.Bottema, O. "De stelling van Pompeïu." Nieuw Tijdschr. Wisk. 44, 183-184, 1956-1957.Bottema, O. "On a Relation Between Four Line Segments." Annali di Math. 70, 295-304, 1965.Bottema, O. Verscheidenheden. Groningen, Netherlands: Wolters-Noordhoff/NVvW, pp. 134-137, 1977.Pompeïu, D. "Une identité entre nombres complexes et un théorème du géometrie élémentaire." Bull. Math. Phys. École Polytechnique Bucarest 6, 6-7, 1936.Pavlovic, S. V. "Sur un démonstration géométrique d'un théorème de M. D. Pompeïu." Elem. Math. 8, 13-15, 1953.Pavlovic, S. V. "Ueber die Erweiterung eines elementargeometrischen Satzes von D. Pompeïu." Abh. Math. Sem. Hamburg 30, 54-60, 1967.Sydler, J. P. "Autre démonstration du théorème de Pompeïu." Elem. Math. 8, 15-16, 1953.Veldkamp, G. R. "Een stelling uit de elementaire meetkunde van het platte vlak." Nieuw Tijdschr. Wisk. 44, 1-4, 1956-1957.Veldkamp, G. R. "Nog een generalisatie van de stelling van Pompeïu." Nieuw Tijdschr. Wisk. 45, 197-204, 1957-1958.

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Pompeiu's Theorem

Cite this as:

van Lamoen, Floor. "Pompeiu's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PompeiusTheorem.html

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