TOPICS
Search

Inner Napoleon Triangle

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
InnerNapoleonTriangle

The inner Napoleon triangle is the triangle DeltaN_AN_BN_C formed by the centers of internally erected equilateral triangles DeltaABE_C, DeltaACE_B, and DeltaBCE_A on the sides of a given triangle DeltaABC. By Napoleon's theorem, it is an equilateral triangle. It has trilinear vertex matrix

[1 2sin(C-1/6pi) 2sin(B-1/6pi); 2sin(C-1/6pi) 1 2sin(A-1/6pi); 2sin(B-1/6pi) 2sin(A-1/6pi) 1]

(Kimberling 1998, p. 171; typo corrected) and area

Delta^'=1/2Delta-1/(24)sqrt(3)(a^2+b^2+c^2),

where Delta is the area of the original triangle.

The circumcircle of the inner Napoleon triangle is the inner Napoleon circle.

All triangle centers of the inner Napoleon triangle correspond to the triangle centroid of the reference triangle.

See also

Inner Napoleon Circle, Outer Napoleon Triangle

Explore with Wolfram|Alpha

References

Belenkiy, I. "New Features of Napoleon's Triangles." J. Geom. 66, 17-26, 1999.Coxeter, H. S. M. and Greitzer, S. L. "Napoleon Triangles." §3.3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Rigby, J. F. "Napoleon Revisited." J. Geom. 33, 129-146, 1988.Yaglom, I. M. Geometric Transformations I. New York: Random House, pp. 38 and 93, 1962.

Referenced on Wolfram|Alpha

Inner Napoleon Triangle

Cite this as:

Weisstein, Eric W. "Inner Napoleon Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InnerNapoleonTriangle.html

Subject classifications