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Graves's Theorem

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GravessTheorem

Graves's theorem states that if a closed nonstretchable loop of string longer than the perimeter of a nondegenerate ellipse is pulled taut around the ellipse by a point in its exterior, then the locus of that point is an ellipse confocal with the original ellipse (F. van Lamoen, pers. comm., Oct. 11, 2006; Tabachnikov 2023). A circle gives a larger circle with the same center, corresponding to coincident foci.

See also

Confocal Ellipses, Ellipse

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References

Poorrezaei, K. "Two Proofs of Graves's Theorem." Amer. Math. Monthly 110, 826-830, 2003. https://doi.org/10.2307/3647801.Tabachnikov, S. "Billiards in Conics Revisited." Stony Brook University, Oct. 2023. https://www.math.stonybrook.edu/PDFs/Sergei_Tabachnikov.pdf.

Cite this as:

Weisstein, Eric W. "Graves's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GravessTheorem.html

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