Napoleon ’s theorem is the statement of Euclidean planimetry on equilateral triangles:
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Triangles can be built inward (all) - the statement will remain valid.
The triangle thus obtained is called the Napoleon triangle (internal and external).
The theorem is often attributed to Napoleon Bonaparte (1769-1821). It is possible, however, that it was proposed by W. Rutherford in the 1825 publication The Ladies' Diary . [one]
Content
Evidence
This theorem can be proved in several ways. One of them uses the rotation and the Schal theorem (3 consecutive rotations return the plane to its place). A similar method uses rotary homothety (when applying 2 homotheties with equal coefficients, MN and LN go into one segment CZ). Other methods are more straightforward, but also more cumbersome and complex.
Napoleon Center
See also Napoleon's Points .
The figure for the paragraph is located at: http://faculty.evansville.edu/ck6/tcenters/class/xsub17.gif Let triangle ABC be given and let D, E, F be points in the figure for which triangles DBC, CAE, ABF equilateral. Further, let: G be the center of the triangle DBC , H the center of the triangle CAE , I the center of the triangle ABF . Then the segments AG, BH, CI intersect at one point. Denote this point by the letter N. This is the so-called first Napoleon point. The trilinear coordinates for point N are: csc (A + π / 6): csc (B + π / 6): csc (C + π / 6). If the equilateral triangles DBC, CAE, ABF are not built outward but inside the given triangle ABC , then the three lines AG, BH, CI intersect at the second Napoleon point. Its trilinear coordinates are: csc (A - π / 6): csc (B - π / 6): csc (C - π / 6).
Note
The first and second points of Napoleon in the Encyclopedia of Points of the Triangle of Clark Kimberling (Clark Kimberling. Encyclopedia of Triangle Centers = http://faculty.evansville.edu/ck6/encyclopedia/ ) are known as points X (17) and X (18).
Links
NAPOLEON POINTS. http://faculty.evansville.edu/ck6/tcenters/class/napoleon.html Dao Thanh Oai. Equilateral Triangles and Kiepert Perspectors in Complex Numbers, Forum Geometricorum 15 (2015) 105-114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html Dao Thanh Oai. A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette 99 (March 2015) 151-153. http://journals.cambridge.org/action/displayIssue?jid=MAG&volumeId=99&seriesId=0&issueId=544 John Rigby. "Napoleon revisited," Journal of Geometry 33 (1988) 129-146. Encyclopedia of Triangle Centers. X (17) and X (18). http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
Relationship with other statements
- Generalization - the Peter-Neumann-Douglas theorem [2]
Napoleon's theorem is generalized to the case of arbitrary triangles as follows:
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An analogue of Napoleon's theorem for parallelograms is the first Tebo theorem .
See also
- Wonderful Triangle Points
- Triangle geometry
- Right triangle
- Farm Second Point
- Van Aubel's Theorem
- Thebo Theorem 2 and 3
- Farm Point
- Apollonia points
- Napoleon's points
- Torricelli points
- Triangle
- Segments and circles associated with a triangle
Links
- Napoleon's theorem in animation.
- Weisstein, Eric W. Napoleon's theorem on the Wolfram MathWorld website.