The Napoleon Triangles

... and Interessting Relationships

 

 by

  Markus Heisss

  Würzburg, Bavaria

  2018/2019/2020/2022/2024

Last Update: June 7, 2024

 

    The copying of the following graphics is allowed, but without changes.

 [To get a bigger picture, please click it with the cursor.]

 

 

  Let's start with the outer Napoleon triangle :

 

Napoleon triangles, named after the famous emperor; Graphic by Heisss, Germany
Fig. 01: The Outer Napoleon Triangle and Outer Napoleon Circle

 

And now the inner Napoleon triangle as graphic:

 

Napoleon triangles, Graphic by Heisss, Germany
Fig. 02: The Inner Napoleon Triangle and Inner Napoleon Circle

 

These triangles are named after the French Emperor Napoleon Bonaparte!

If he discovered these geometric relationships by himself is uncertain.

 

Further information you get from the internet under:

[mathworld.wolfram.com]

 

Connection of both Napoleon triangles

with the McCay circles and the circle of Apollonius:

 

 

Now both cases from above together, but first only at side AB:

 

Apollonius circle, McCay circles, Napoleon triangles, Graphic by Heisss
Fig. 03: Both equilateral triangles at side AB and Apollonius circle as c-McCay circle

 

Further information to "Circle of Apollonius": [here]

 

And finally the same at all three sides of the triangle:

 

Napoleon triangles, McCay circles, Graphic by Heisss, Germany, Art Painter and Geometrician
Fig. 04: Vertices of the Napoleon triangles and the McCay circles

 

[To the Proof]

 

Further information to "McCay circles": [here]

 

 

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Postscript, March 24, 2022:

 

Relationship of the Napoleon circles

with the Steiner circumellipse:

 

Napoleon, circle, triangle, Steiner, circumellipse
Fig. 05: Relationship between the two Napoleon circles and the Steiner circumellipse

 

Please note also the simple relationships between the radii of the Napoleon circles

and the semi-axes of the Steiner circumellipse!

 

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Postscript, May 31, 2024:

 

... And further discoveries to the Napoleon triangles:

 

first Napoleon point, Kimberling center X17
Fig. 06: Medial triangle of the inner Napoleon triangle

 

Analogue with the inner Napoleon triangle:

 

second Napoleon point, Napoleon circles
Fig. 07: Medial triangle of the outer Napoleon triangle

 

And now a relationship with radical lines:

 

radical axis, radical center
Fig. 08: The vertices of the inner Napoleon triangle lie on a radical line

 

Again, analogue with the inner Napoleon triangle:

 

radical axis, Heisss, Markus Heiss, Geometry, Würzburg,
Fig. 09: The vertices of the outer Napoleon triangle lie on a radical line

 

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Postscript, March 23, 2020:

 

... And further discoveries to the Napoleon triangles:

 

Napoleon triangle, coaxal system, Geometry, Heisss
Fig. 10: Vertices of the Napoleon triangles and coaxal circles
And another connection between

the Napoleon triangles and the Steiner circumellipse:

Napoleon triangle, Steiner circumellipse
Fig. 11: Vertices of Napoleon triangles and collinear incenters

 

More information about the Steiner circumellipse? [here]
... And another discovery:
Napoleon triangle, Euler line, Geometry
Fig. 12: Vertices of the Napoleon triangles and three centroids

 

... and now triangles with the nine-point center:
Nine-point circle, Napoleon triangle, centroid, Geometry, Heisss,
Fig. 13: Nine-point enter, Napoleon triangles and three centroids

 

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Are you interested in my other geometrical discoveries?

[here]