As perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas.
12.
The sum of these areas will always be greater than the area without discrimination assuming the demand curve resembles a rectangular hyperbola with unitary elasticity.
13.
In particular when we choose the rectangular hyperbola " xy " = 1, one recovers the " natural logarithm ".
14.
Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless ? = 45?( a rectangular hyperbola ).
15.
The well documented rectangular hyperbolas that pass through four orthocentric points are the Feuerbach, JeY�bek and Kiepert circumhyperbolas of the reference triangle ABC in a normalized system with H as the orthocenter.
16.
This conic is a rectangular hyperbola and it is called the "'Kiepert hyperbola "'in honor of Ludwig Kiepert ( 1846 1934 ), the mathematician who discovered this result.
17.
Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.
18.
*Dao's theorem on a rectangular hyperbola, the this article for this section : " Dao Thanh, Oai ( 2014 ) . " A Simple Proof of Gibert s Generalization of the Lester Circle Theorem ".
19.
So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles.
20.
Christopher Bath describes a nine-point rectangular hyperbola passing through these centers : incenter X ( 1 ), the three excenters, the centroid X ( 2 ), the de Longchamps point X ( 20 ), and the three points obtained by extending the triangle medians to twice their cevian length.
