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Proportionally-Cutting Circle


Proportionally cutting circles are circles that intersect the sidelines of a reference triangle DeltaABC such that length of the chords that are cut off have lengths mua, mub, and muc that are proportional to the corresponding side lengths a , b, and c of DeltaABC.

The circumcircle and Stammler circles are proportionally-cutting circles. Let O_A be the A-vertex of the anticevian triangle of the circumcenter O, then the circle with center O_A passing through A is a proportionally cutting circle, and similarly for O_B and O_C.

The centers of proportionally-cutting circles lie on the Stammler hyperbola.

Depending on mu, there may be 2, 3 or 4 proportionally-cutting circles (Stammler 1992). If mu is a value giving four proportionally-cutting circles, then the centers of these circles form an orthocentric system of which the circumcircle is the nine-point circle (Ehrmann and van Lamoen 2002).


See also

Circumcircle, Orthocentric System, Stammler Circles, Stammler Hyperbola, Stammler Triangle

This entry contributed by Floor van Lamoen

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References

Ehrmann, J.-P. and van Lamoen, F. M. "The Stammler Circles." Forum Geom. 2, 151-161, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200219index.html.Stammler, L. "Der Kreis mit minimaler Flächendifferenz zum Dreieck. Elem. Math. 33, 143-152, 1978.Stammler, L. "Optimierung konstant belegter Ovalen und ein Hyperbel-Schnittzats." Beitr. Alg. Geom., 29, 21-34, 1989.Stammler, L. "Proportionalschnittkreise, ihre Mittenhyperbel und ein Pendant zum Satz von Morley." Elem. Math. 47, 148-158, 1992.Stammler, L. "Cutting Circles and the Morley Theorem." Beitr. Alg. Geom. 38, 91-93, 1997. http://www.emis.de/journals/BAG/vol.38/no.1/7.html.

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Proportionally-Cutting Circle

Cite this as:

van Lamoen, Floor. "Proportionally-Cutting Circle." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Proportionally-CuttingCircle.html

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