Hyperbol Cyprus

Point on Cybert's hyperbole.

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

Hyperbole of Cypert is a hyperbole defined by this triangle . If the latter is a triangle in general position, then this hyperbola is the only conical section passing through its vertices, the orthocenter and the centroid .

Content

1 Definition via isogonal pairing
2 Definition through triangles in trilinear coordinates
3 Trilinear coordinates of an arbitrary point N lying on the Cybert hyperbole
4 The equation of Cybert hyperbole in trilinear coordinates
5 Known points lying on Cybert's hyperbole
6 List of points lying on Kipert's hyperbole
7 Generalization of the Leicester theorem in the form of the theorem of B. Gibert (2000)
8 History
9 Properties
10 See also
11 Notes
12 Literature

Isogonal pairing

Kipert’s hyperbola is a curve isogonally conjugate to a straight line passing through the Lemoine point and the center of the circumscribed circle of a given triangle.

A straight line passing through the center of the circumscribed circle and the point of Lemoine is called the Brocard axis . On it lie the points of Apollonius . In other words, Kipert’s hyperbola is a curve isogonally conjugate to the Brocard axis of a given triangle.

Definition through triangles in trilinear coordinates

Definition through triangles in trilinear coordinates ^[1] :

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 equally spaced (that is, they are all built either from the outside or from 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 an arbitrary point N lying on the Cybert hyperbole

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

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

Kipert hyperbole equation in trilinear coordinates

The geometric 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 Kipert’s hyperbola with the equation

{\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.

Known points lying on Kipert's hyperbole

Among the points lying on the Kipert hyperbole, there are such important points of the triangle ^[2] :

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)
$\mathrm {arctg} \,[\mathrm {tg} \,(A/2)\mathrm {tg} \,(B/2)\mathrm {tg} \,(C/2)]$ ${\ displaystyle \ mathrm {arctg} \, [\ mathrm {tg} \, (A / 2) \ mathrm {tg} \, (B / 2) \ mathrm {tg} \, (C / 2)]}$ ^[3]	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

The list of points lying on Kipert's hyperbole

Kipert’s hyperbola passes through the following centers of the triangle X (i) ^[3] :

for i = 2, ( Centroid of the triangle ),
i = 4 ( Orthocenter ),
i = 10 ( Spiker Center ; that is, the incenter of a triangle with vertices in the middle of the sides of a given triangle ABC ^[1] ),
i = 13 (first Fermat point ), i = 14 (second Fermat point ),
i = 17 ( first point of Napoleon ), i = 18 ( second point of Napoleon ),
i = 76 (third point of Brokar ),
i = 83 (the point isogonally conjugate to the midpoint between the points of Brokar ^[1] ),
i = 94, 96,
i = 98 ( Tarry point)
i = 226, 262, 275, 321,
i = 485 ( Outside point of Vekten ), i = 486 ( Outside point of Vekten ),
i = 598, 671, 801, 1029, 1131, 1132,
i = 1139 (inner pentagon point = inner pentagon point), i = 1140 (outer pentagon point = outer pentagon point),
i = 1327, 1328, 1446, 1676, 1677, 1751, 1916, 2009, 2010, 2051, 2052, 2394, 2592, 2593,
i = 2671 (first golden arbelos point = first golden arbelos point),
i = 2672 (second golden arbelos point = second golden arbelos point),
i = 2986, 2996

A generalization of the Leicester theorem in the form of the theorem of B. Gibert (2000)

B. Hibert’s theorem (2000) generalizes the Leicester circle theorem, namely: any circle whose diameter is the chord of the Kipert hyperbola of a triangle and is perpendicular to its Euler line passes through the Fermat points ^[4] ^[5] .

History

This hyperbole was named in honor of the German mathematician Friedrich Wilhelm August Ludwig Kiepert who discovered it (Friedrich Wilhelm August Ludwig Kiepert, 1846-1934) ^[1] .

Properties

Kipert’s hyperbola is equilateral (that is, its diagonals are perpendicular), therefore, its center, designated in the encyclopedia of triangle centers as X (115), lies on Euler’s circle .

Notes

↑ ¹ ² ³ ⁴ Eddy, Fritsch, 1994 , p. 188-205.
↑ Hakobyan A.V. , Zaslavsky A.A. Geometric properties of second-order curves. - 2nd ed., Supplemented. - 2011. - S. 125-126.
↑ ¹ ² Weisstein, Eric W. Kiepert Hyperbola on the Wolfram MathWorld website.
↑ B. Gibert (2000): [Message 1270] . Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
↑ Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations . Forum Geometricorum, volume 10, pages 175-209. MR : 2868943

Literature

Eddy R. H., Fritsch R. The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle // Math Magazine , 1994, 67 . - P. 188-205.

[_2a7371577f47cf63-1] ¹ ² ³ ⁴ Eddy, Fritsch, 1994 , p. 188-205.

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

[autogenerated1-3] ¹ ² Weisstein, Eric W. Kiepert Hyperbola on the Wolfram MathWorld website.

[4] B. Gibert (2000): [Message 1270] . Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.

[5] Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations . Forum Geometricorum, volume 10, pages 175-209. MR : 2868943