Feuerbach hyperbola explained

In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Schiffler point. The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle.[1]

Equation

\cosB-\cosC
\alpha

+

\cosC-\cosA
\beta

+

\cosA-\cosB
\gamma
(here

A,B,C

are the angles at the respective vertices and

(\alpha,\beta,\gamma)

is the barycentric coordinate).[2]

Properties

Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle

ABC

lies on the curve.

The line

OI

is tangent to this hyperbola at

I

.

Isogonal conjugate of OI

The hyperbola is the isogonal conjugate of

OI

, the line joining the circumcenter and the incenter.[3] This fact leads to a few interesting properties. Specifically all the points lying on the line

OI

have their isogonal conjugates lying on the hyperbola. The Nagel point lies on the curve since its isogonal conjugate is the point of concurrency of the lines joining the vertices and the opposite Mixtilinear incircle touchpoints, also the in-similitude of the incircle and the circumcircle. Similarly, the Gergonne point lies on the curve since its isogonal conjugate is the ex-similitude of the incircle and the circumcircle.

The pedal circle of any point on the hyperbola passes through the Feuerbach point, the center of the hyperbola.

Kariya's theorem

Given a triangle

ABC

, let

A1,B1,C1

be the touchpoints of the incircle

\odotI

with the sides of the triangle opposite to vertices

A,B,C

respectively. Let

X,Y,Z

be points lying on the lines

IA1,IB1,IC1

such that

IX=IY=IZ

. Then, the lines

AX,BY,CZ

are concurrent at a point lying on the Feuerbach hyperbola.

The Kariya's theorem has a long history.[4] It was proved independently by Auguste Boutin and V. Retali.,[5] [6] [7] but it became famous only after Kariya's paper.[8] Around that time, many generalizations of this result were given. Kariya's theorem can be used for the construction of the Feuerbach hyperbola.

Both Lemoine's theorem and Kariya's theorem are a special case of Jacobi's theorem.

See also

Other rectangular hyperbolas

References

  1. Boucher. H.. 1893. Essai de classification sur les races gallines. Annales de la Société linnéenne de Lyon. 40. 1. 89–100. 10.3406/linly.1893.4047. 1160-6398.
  2. Parry. C. F.. 2001. Triangle centers and central triangles, by Clark Kimberling (Congress Numerantium Vol. 129) Pp. 295. $42.50 1998. 0316-1282 (Utilitas Mathematica Publishing, Inc., Winnipeg).. The Mathematical Gazette. 85. 502. 172–173. 10.2307/3620531. 3620531. 227212286 . 0025-5572.
  3. Rigby. J. F.. 1973. A concentrated dose of old-fashioned geometry. The Mathematical Gazette. 57. 402. 296–298. 10.2307/3616051. 3616051. 126241645 . 0025-5572.
  4. 2012. Problems and Solutions. The American Mathematical Monthly. 119. 8. 699. 10.4169/amer.math.monthly.119.08.699. 37903933.
  5. Kahane. J.. 1961. Problèmes et remarques sur les carrés de convolution. Colloquium Mathematicum. 8. 2. 263–265. 10.4064/cm-8-2-263-265. 0010-1354. free.
  6. Humbert. G.. 1890. Sur les coniques inscrites à une quartique. Annales de la faculté des sciences de Toulouse Mathématiques. 4. 3. 1–8. 10.5802/afst.55. 0996-0481. free.
  7. 1889. Periodico di Matematica per ľinsegnamento secondario. Rendiconti del Circolo Matematico di Palermo. 3. 2. 56. 10.1007/bf03017173. 184480136. 0009-725X.
  8. Kariya. J.. 1904. Un probleme sur le triangle. L'Enseignement Mathématiques. 6. 130–132, 236, 406.

Further reading