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Outer Napoleon Triangle


OuterNapoleonTriangle

The outer Napoleon triangle is the triangle DeltaN_C^'N_B^'N_A^' formed by the centers of externally erected equilateral triangles DeltaABE_C^', DeltaACE_B^', and DeltaBCE_A^' on the sides of a given triangle DeltaABC. By Napoleon's theorem, it is also 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; typos 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 lines connecting the vertices of the outer Napoleon triangle with the opposite vertices of the original triangle concur in the first Napoleon point, and similarly for the inner Napoleon triangle, where the corresponding lines concur in the second Napoleon point.

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

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


See also

Equilateral Triangle, First Napoleon Point, Inner Napoleon Triangle, Inner Vecten Triangle, Kiepert Hyperbola, Lemoine's Problem, Napoleon Points, Napoleon's Theorem, Outer Napoleon Circle, Second Napoleon Point

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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.

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Outer Napoleon Triangle

Cite this as:

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

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