Steiner Deltoid
The Steiner deltoid is the envelope of the Simson lines of a triangle.
Its circumcircle is the Steiner circle, and its incircle is the nine-point circle.
The triangle formed by vertices of the deltoid is homothetic to the first Morley triangle, with the center dividing the line X_5X_(356) in the ratio of their side lengths,
| k=3/(16)sqrt(3)csc(1/3A)csc(1/3B)csc(1/3C). |
The triangle formed by the deltoid's meets with the nine-point circle is also is homothetic to the first Morley triangle, with the center dividing the line X_5X_(356) in the ratio of their side lengths
| k=-1/(16)sqrt(3)csc(1/3A)csc(1/3B)csc(1/3C). |
See also
Deltoid, First Morley Triangle, Simson Line, Steiner CircleThis entry contributed by Peter Moses
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References
Butchart, J. H. "The Deltoid Regarded as the Envelope of Simson Lines." Amer. Math. Monthly 46, 85-86, 1939.de Guzmán, M. "The Envelope of the Wallace-Simson Lines of a Triangle: A Simple Proof of the Steiner Theorem on the Deltoid." Dec. 1998. http://www.mat.ucm.es/deptos/am/guzman/deltoide121298/00delten.htm.Dörrie, H. "Steiner's Three-Pointed Hypocycloid." §53 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 226-231, 1965.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1967.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 155 and 230-231, 1991.Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover, 1963.Referenced on Wolfram|Alpha
Steiner DeltoidCite this as:
Moses, Peter. "Steiner Deltoid." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SteinerDeltoid.html