Napoleon point

The two Napoleon points , named after the French general and Emperor Napoléon Bonaparte , are among the excellent points in the triangle .

The 1st Napoleon point is defined as follows:

Three equilateral triangles are drawn outward over the sides of a given triangle . If you connect the centers of gravity of these triangles with the opposite corners of the original triangle, the connecting straight lines intersect at a point, the 1st Napoleon point of the given triangle.

If you draw the equilateral triangles on the other side, you get the 2nd Napoleon point accordingly .

The connecting lines of the three focal points always form an equilateral triangle, regardless of the length of the base of the touchdown triangle.

properties

• The two Napoleon points lie on the Kiepert hyperbola .

Coordinates

Napoleon points ( and ) ${\displaystyle X_{17}}$ ${\displaystyle X_{18}}$
Trilinear coordinates	${\displaystyle \csc \left(\alpha \pm {\frac {\pi }{6}}\right)\,:\,\csc \left(\beta \pm {\frac {\pi }{6}}\right)\,:\,\csc \left(\gamma \pm {\frac {\pi }{6}}\right)}$
Barycentric coordinates	${\displaystyle a\cdot \csc \left(\alpha \pm {\frac {\pi }{6}}\right)\,:\,b\cdot \csc \left(\beta \pm {\frac {\pi }{6}}\right)\,:\,c\cdot \csc \left(\gamma \pm {\frac {\pi }{6}}\right)}$

See also

Excellent points in the triangle , Napoleon triangle

Web links

• Eric W. Weisstein : Napoleon Points . In: MathWorld (English).
• Napoleon point - a visualization of the 1st Napoleon point with the dynamic geometry program GeoGebra