Napoleon's points

Napoleon points in geometry - a pair of special points on the plane of a triangle . The legend attributes the discovery of these points to the French Emperor Napoleon I , however his authorship is doubtful ^[1] . Napoleon's points belong to the remarkable points of the triangle and are listed in the Encyclopedia of the centers of the triangle as points X (17) and X (18).

The name “Napoleon points” also applies to various pairs of centers of the triangle, better known as isodynamic points ^[2] .

Point Definition

Napoleon's First Point

The first point of Napoleon

Let ABC be any triangle in the plane . On the sides BC , CA , AB of the triangle we construct the external regular triangles DBC , ECA and FAB, respectively. Let the centroids of these triangles be X , Y, and Z, respectively. Then the straight lines AX , BY and CZ intersect at one point, and this point N1 is the first (or outer) Napoleon point of triangle ABC .

The triangle XYZ is called the external Napoleon triangle of triangle ABC . Napoleon's theorem states that this triangle is regular .

In the Encyclopedia of Triangle Centers, the first point of Napoleon is designated as X (17). ^[3]

Trilinear coordinates of point N1:

{\begin{aligned}&\left(\csc \left(A+{\frac {\pi }{6}}\right),\csc \left(B+{\frac {\pi }{6}}\right),\csc \left(C+{\frac {\pi }{6}}\right)\right)\\&=\left(\sec \left(A-{\frac {\pi }{3}}\right),\sec \left(B-{\frac {\pi }{3}}\right),\sec \left(C-{\frac {\pi }{3}}\right)\right)\end{aligned}}

{\ displaystyle {\ begin {aligned} & \ left (\ csc \ left (A + {\ frac {\ pi} {6}} \ right), \ csc \ left (B + {\ frac {\ pi} {6} } \ right), \ csc \ left (C + {\ frac {\ pi} {6}} \ right) \ right) \\ & = \ left (\ sec \ left (A - {\ frac {\ pi} { 3}} \ right), \ sec \ left (B - {\ frac {\ pi} {3}} \ right), \ sec \ left (C - {\ frac {\ pi} {3}} \ right) \ right) \ end {aligned}}}

The barycentric coordinates of point N1:

\left(a\csc \left(A+{\frac {\pi }{6}}\right),b\csc \left(B+{\frac {\pi }{6}}\right),c\csc \left(C+{\frac {\pi }{6}}\right)\right)

{\ displaystyle \ left (a \ csc \ left (A + {\ frac {\ pi} {6}} \ right), b \ csc \ left (B + {\ frac {\ pi} {6}} \ right), c \ csc \ left (C + {\ frac {\ pi} {6}} \ right) \ right)}

Napoleon's Second Point

The second point of Napoleon

Let ABC be any triangle in the plane . On the sides BC , CA , AB of the triangle we construct the inner equilateral triangles DBC , ECA and FAB, respectively. Let X , Y, and Z be the centroids of these triangles, respectively. Then the lines AX , BY and CZ intersect at one point, and this point N2 is the second (or internal) Napoleon point of triangle ABC .

The triangle XYZ is called the inner Napoleon triangle of triangle ABC . Napoleon's theorem states that this triangle is regular .

In the Encyclopedia of Triangle Centers, the second point of Napoleon is designated as X (18). ^[3]

Trilinear coordinates of point N2:

{\begin{aligned}&\left(\csc \left(A-{\frac {\pi }{6}}\right),\csc \left(B-{\frac {\pi }{6}}\right),\csc \left(C-{\frac {\pi }{6}}\right)\right)\\&=\left(\sec \left(A+{\frac {\pi }{3}}\right),\sec \left(B+{\frac {\pi }{3}}\right),\sec \left(C+{\frac {\pi }{3}}\right)\right)\end{aligned}}

{\ displaystyle {\ begin {aligned} & \ left (\ csc \ left (A - {\ frac {\ pi} {6}} \ right), \ csc \ left (B - {\ frac {\ pi} { 6}} \ right), \ csc \ left (C - {\ frac {\ pi} {6}} \ right) \ right) \\ & = \ left (\ sec \ left (A + {\ frac {\ pi } {3}} \ right), \ sec \ left (B + {\ frac {\ pi} {3}} \ right), \ sec \ left (C + {\ frac {\ pi} {3}} \ right) \ right) \ end {aligned}}}

The barycentric coordinates of point N2:

\left(a\csc \left(A-{\frac {\pi }{6}}\right),b\csc \left(B-{\frac {\pi }{6}}\right),c\csc \left(C-{\frac {\pi }{6}}\right)\right)

{\ displaystyle \ left (a \ csc \ left (A - {\ frac {\ pi} {6}} \ right), b \ csc \ left (B - {\ frac {\ pi} {6}} \ right ), c \ csc \ left (C - {\ frac {\ pi} {6}} \ right) \ right)}

Two points closely related to Napoleon's points are Fermat's points (X13 and X14 in the encyclopedia of points). If instead of the straight lines connecting the centroids of equilateral triangles with the corresponding vertices, draw straight lines connecting the vertices of equilateral triangles with the corresponding vertices of the original triangle, so the three lines built will intersect at one point. Intersection points are called Fermat points and are designated as F1 and F2. The intersection of the Fermat line (that is, the line connecting the two points of Fermat) and the line of Napoleon (that is, the line connecting the two points of Napoleon) is a triangle simedian (point X6 in the encyclopedia of centers).

Properties

Hyperbol Cyprus

Cypert's hyperbola is the described hyperbola passing through the centroid and orthocenter . If we construct similar isosceles triangles (outward or inward) on the sides of the triangle, and then connect their vertices with the opposite vertices of the original triangle, then three such lines intersect at one point lying on the Cybert hyperbole. In particular, this hyperbole contains Torricelli points and Napoleon points (Chevian intersection points connecting the vertices to the centers of regular triangles built on opposite sides) ^[4] .

Generalizations

The result of the existence of Napoleon's points can be generalized in various ways. When determining Napoleon's points, we used equilateral triangles built on the sides of the triangle ABC, and then we selected the centers X, Y, and Z of these triangles. These centers can be considered as the vertices of isosceles triangles , built on the sides of the triangle ABC with an angle at the base of π / 6 (30 degrees). Generalizations consider other triangles that, being built on the sides of the triangle ABC, have similar properties, that is, the lines connecting the vertices of the constructed triangles with the corresponding vertices of the original triangle intersect at one point.

Isosceles Triangles

Point on Cybert's hyperbole.

Hyperport of Kipert triangle ABC. The hyperbola passes through the vertices (A, B, C), the orthocenter (O) and the centroid (G) of the triangle.

This generalization states: ^[5]

If the three triangles XBC, YCA and ZAB are built on the sides of the triangle ABC, are similar , isosceles with bases on the sides of the original triangle, and are equally spaced (that is, they are all built on the outside, or all are built on the inside), then the lines AX, BY and CZ intersect at one point N.

If the total angle at the base is $\theta$ ${\ displaystyle \ theta}$ , then the vertices of three triangles have the following trilinear coordinates.

$X(-\sin \theta ,\sin(C+\theta ),\sin(B+\theta ))$ ${\ displaystyle X (- \ sin \ theta, \ sin (C + \ theta), \ sin (B + \ theta))}$
$Y(\sin(C+\theta ),-\sin \theta ,\sin(A+\theta ))$ ${\ displaystyle Y (\ sin (C + \ theta), - \ sin \ theta, \ sin (A + \ theta))}$
$Z(\sin(B+\theta ),\sin(A+\theta ),-\sin \theta )$ ${\ displaystyle Z (\ sin (B + \ theta), \ sin (A + \ theta), - \ sin \ theta)}$

Trilinear coordinates of point N

(\csc(A+\theta ),\csc(B+\theta ),\csc(C+\theta ))

{\ displaystyle (\ csc (A + \ theta), \ csc (B + \ theta), \ csc (C + \ theta))}

A few special cases.

Value $\theta$ ${\ displaystyle \ theta}$	Point $N$ ${\ displaystyle N}$
0	G, centroid of triangle ABC (X2)
π / 2 (or, - π / 2)	O, the orthocenter of triangle ABC (X4)
$\mathrm {arctg} \,[\mathrm {tg} \,(A/2)\mathrm {tg} \,(B/2)\mathrm {tg} \,(C/2)]$ ${\ displaystyle \ mathrm {arctg} \, [\ mathrm {tg} \, (A / 2) \ mathrm {tg} \, (B / 2) \ mathrm {tg} \, (C / 2)]}$ ^[6]	Speaker Center (X10)
π / 4	Vecten points (X485)
- π / 4	Vecten Points (X486)
π / 6	N1, the first point of Napoleon (X17)
- π / 6	N2, second point of Napoleon (X18)
π / 3	F1, Fermat's first point (X13)
- π / 3	F2, Fermat's second point (X14)
- A (if A <π / 2) π - A (if A> π / 2)	Top a
- B (if B <π / 2) π - B (if B> π / 2)	Top B
- C (if C <π / 2) π - C (if C> π / 2)	Vertex c

Moreover, the geometrical location of the points N when changing the angle at the base of the triangles $\theta$ ${\ displaystyle \ theta}$ between -π / 2 and π / 2 is a hyperbole

{\frac {\sin(B-C)}{x}}+{\frac {\sin(C-A)}{y}}+{\frac {\sin(A-B)}{z}}=0,

{\ displaystyle {\ frac {\ sin (BC)} {x}} + {\ frac {\ sin (CA)} {y}} + {\ frac {\ sin (AB)} {z}} = 0, }

Where $x, y, z$ ${\ displaystyle x, y, z}$ - trilinear coordinates of point N in a triangle.

History

This hyperbola is called the Kipert hyperbole (in honor of the German mathematician Friedrich Wilhelm August Ludwig Kiepert who discovered it) (Friedrich Wilhelm August Ludwig Kiepert), 1846-1934 ^[5] . This hyperbola is the only conical section passing through points A, B, C, G and O.

Note

The Speaker Center has a very similar property. Shpiker's center S is the intersection point of lines AX , BY and CZ , where the triangles XBC , YCA and ZAB are similar, isosceles and equally spaced, built on the sides of the triangle ABC from the outside, having the same angle at the base $\mathrm {arctg} \,[\mathrm {tg} \,(A/2)\mathrm {tg} \,(B/2)\mathrm {tg} \,(C/2)]$ ${\ displaystyle \ mathrm {arctg} \, [\ mathrm {tg} \, (A / 2) \ mathrm {tg} \, (B / 2) \ mathrm {tg} \, (C / 2)]}$ ^[6] .

Similar triangles

Generalization of Napoleon's point - a special case

For the three lines AX, BY, and CZ to intersect at one point, the triangles XBC, YCA, and ZAB, built on the sides of the triangle ABC, do not have to be isosceles ^[7] .

If similar triangles XBC, AYC and ABZ are built from the outside on the sides of an arbitrary triangle ABC, then the lines AX, BY and CZ intersect at one point.

Arbitrary Triangles

Lines AX, BY and CZ intersect at one point even under milder conditions. The following condition is one of the most general conditions for lines AX, BY, and CZ to intersect at one point ^[7] .

If the triangles XBC, YCA, and ZAB are constructed externally on the sides of the triangle ABC so that

∠CBX = ∠ABZ, ∠ACY = ∠BCX, ∠BAZ = ∠CAY,

then lines AX, BY and CZ intersect at one point.

Generalization of Napoleon's point

On the discovery of Napoleon's points

Coxeter and Greitzer formulate Napoleon's theorem as follows: If equilateral triangles are built from the outside on the sides of any triangle, then their centers form an equilateral triangle . They notice that Napoleon Bonaparte was a bit of a mathematician and had a great interest in geometry, but they doubt that he was geometrically formed enough to discover the theorem attributed to him ^[1] .

The earliest extant publication with dots is an article in the annual journal The Ladies' Diary (Women's Diary, 1704-1841) in its 1825 issue. The theorem was an answer to a question sent by W. Rosenford, but Napoleon is not mentioned in this publication.

In 1981, the German mathematician historian Christoph J. Scriba published the results of a study of the assignment of points to Napoleon in the journal Historia Mathematica ^[8] .

Notes

↑ ¹ ² Coxeter, Greitzer, 1967 , p. 61–64.
↑ Rigby, 1988 , p. 129–146.
↑ ¹ ² Kimberling, Clark Encyclopedia of Triangle Centers (neopr.) . Date of treatment May 2, 2012.
↑ Hakobyan A.V. , Zaslavsky A.A. Geometric properties of second-order curves. - 2nd ed., Supplemented. - 2011. - S. 125-126.
↑ ¹ ² Eddy, Fritsch, 1994 , p. 188–205.
↑ ¹ ² Weisstein, Eric W. Kiepert Hyperbola on the Wolfram MathWorld website.
↑ ¹ ² de Villiers, 2009 , p. 138-140.
↑ Scriba, 1981 , p. 458–459.

Literature

JF Rigby. Napoleon revisited // Journal of Geometry. - 1988.- T. 33 , no. 1-2 . - S. 129-146 . - DOI : 10.1007 / BF01230612 .
The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle // Mathematics Magazine. - 1994. - T. 67 June , no. 3 . - DOI : 10.2307 / 2690610 .
Michael de Villiers. Some Adventures in Euclidean Geometry. - Dynamic Mathematics Learning, 2009. - ISBN 9780557102952 .
HSM Coxeter, SL Greitzer. Geometry Revisited. - Mathematical Association of America, 1967. Translation: G.S.M. Coxeter, S.L. Greitzer. New meetings with geometry. - Moscow: "Science", 1978. - (Library of the mathematical circle).
Christoph J Scriba. Wie kommt 'Napoleons Satz' zu seinem namen? // Historia Mathematica. - 1981. - T. 8 , no. 4 . - DOI : 10.1016 / 0315-0860 (81) 90054-9 .
Stachel, Hellmuth. Napoleon's Theorem and Generalizations Through Linear Maps (English) // Contributions to Algebra and Geometry: journal. - 2002. - Vol. 43 , no. 2 . - P. 433–444 .
Grünbaum, Branko. A relative of "Napoleon's theorem" (neopr.) // Geombinatorics . - 2001 .-- T. 10 . - S. 116-121 .
Katrien Vandermeulen, et al. Napoleon, a mathematician? (neoprene.). Maths for Europe. Date of appeal April 25, 2012.
Bogomolny, Alexander Napoleon's Theorem (Neopr.) . Cut The Knot! An interactive column using Java applets. Date of appeal April 25, 2012.
Napoleon's Thm and the Napoleon Points (neopr.) . Date of appeal April 24, 2012.
Weisstein, Eric W. Napoleon Points (Neopr.) . From MathWorld — A Wolfram Web Resource. Date of appeal April 24, 2012.
Philip LaFleur Napoleon's Theorem (Neopr.) . Date of treatment April 24, 2012. Archived on September 7, 2012.
Wetzel, John E. Converses of Napoleon's Theorem (Neopr.) (April 1992). Date of treatment April 24, 2012. Archived on April 29, 2014.

[_ebbfa6cdeda4f31c-1] ¹ ² Coxeter, Greitzer, 1967 , p. 61–64.

[_40d944883ac430dc-2] Rigby, 1988 , p. 129–146.

[ETC-3] ¹ ² Kimberling, Clark Encyclopedia of Triangle Centers (neopr.) . Date of treatment May 2, 2012.

[4] Hakobyan A.V. , Zaslavsky A.A. Geometric properties of second-order curves. - 2nd ed., Supplemented. - 2011. - S. 125-126.

[_215d56577a03a1de-5] ¹ ² Eddy, Fritsch, 1994 , p. 188–205.

[autogenerated1-6] ¹ ² Weisstein, Eric W. Kiepert Hyperbola on the Wolfram MathWorld website.

[_25434c23234e8409-7] ¹ ² de Villiers, 2009 , p. 138-140.

[_f39407791731e38a-8] Scriba, 1981 , p. 458–459.