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