In the Field of the Jerabek Asymptotes

This investigation began when I noticed that the asymptotes of the Jerabek hyperbola go through vertices of the MacBeath inconic, the foci of which (H and O) are isogonal and lie on the hyperbola. The axes of the H and D inconics are parallel and perpendicular, respectively, to the Jerabek asymptotes. The perspector and center of both conics lie on the hyperbola. Asymptotes of a circumhyperbola are often unfriendly beasts, algebraically isolated from the great majority of triangle geometry, however on happy occasions the same field extension happens for a number of points and lines, creating an interrelated set of interesting structures. This happens in the case of the Jerabek asymptotes.

Figure: a very informative picture. The Jerabek hyperbola is blue, the MacBeath inconic, whose foci lie on the Jerabek hyperbola, is red. The H and D inconics, whose perspectors and centers lie on the hyperbola, are light and black. They intersect at J, the Jerabek center. The quartile points, which are the intersections of these inconics with the MacBeath inconic, are red. Strong points are blue and fissile points (with 2 versions) green. The axes of the H and D inconics, red and dashed, are perpendicular to eachother and parallel to the Jerabek asymptotes (blue, dashed).

This is such an interesting picture because there are points from three different fields in the above picture, each used differently. The weak, quartile points in red are intersections of the inconics, the points in green are fissile and live in the field of the Jerabek asymptotes, occur in pairs, or strong lines, and the strong points (blue) determining the basic structure of the picture, both in the hyperbola and the Euler line.

The Field of the Jerabek asymptotes

The equation of a line is computed from the points that determine it using only the multiplication and subtraction operations. The line thus lies in the same computational field as the points. However the infinite point on an asymptote is determined as the conjugate of the intersection of a line with the Steiner ellipse or the circumcircle, generally requiring an extension of the computational field in the form of a square root. The asymptote itself lies in this same computational field. Points determined from this asymptote by intersection with other lines or normal triangular operations also lie in the field.

The usual field of computation for triangle points and lines consists of the rational numbers extended by the three sides of the triangle. The incenter, the orthocenter, and most points in the triangle plane are computed in this field. The lines between these points lie in this field, as do the points determined by these lines. If the field is extended by a square root, such as occurs when a circle and line intersect, this intersection will lie on the aforementioned lines and points only under unusual circumstances. This new point is algebraically and geometrically isolated from the others.

However all is not lost for this new point. Standard triangle operations will produce more points of the same type, meaning that they are in the same computational field, which may generate a whole world of computational and geometric relationships of their own. In very special ways they may also touch the points and lines of other fields.

The Jerabek asymptotes and associated points and lines form such a system. Included are:

The Jerabek asymptotes meet the major axis vertices of the MacBeath ellipse, so these vertices lie in the same field as the asymptotes.
The infinite points on the Jerabek asymptotes.
The orthic inconic has axes parallel to the Jerabek asymptotes; sharing the same points at infinity.
The axes of the D-inconic.
The intersections of the Euler line with the circumcircle.
The intersections of the HD line with the Steiner ellipse.
The duals of the above asymptotes and axes are included.


This is quite a collection of lines and points; they both exist in their own world and touch upon the usual geometric world in interesting ways.

Basic properties and results

The Jerabek hyperbola contains the orthocentre and its isotomic conjugate. As such it has irrational asymptotes. The circumconic that switches the Jerabek perspector and center contains the duals of the asymptotes, isotomic and irrational points mentioned above. Its asymptotes are rational, but at the expense of containing these irrational points and the others generated from them. The switched circumconic intersects the duel Jerabek inconic at precisely these irrational points.

The duals of the Jerabek points at infinity are lines through G containing the duals of the axes and asymptotes. These lines are tangent to the dual Jerabek inconic at the point where the inconic meets the switched conic. These points are the duals of the Jerabek asymptotes and are isotomic.

The isotomics of the duals of conic axes are collinear with G on a strong line. The isotomics of the duals of the duals of the center K of the orthic inconic are the Euler intersections with the circumcircle. In other words, the tripoles of the H inconic axes are the Euler intersections with the circumcircle.

Call the asymptotes a1 and a2 which are lines whose coefficients are of the form  p ± q Z

where Z = √(a2b2c2 (–9SABC + S2 SW))

establishing the nature of the irrational term.

The duals of the Jerabek asymptotes
The duals ~a1 and ~a2 (= t ~a1) are the intersections of the Jerabek dual inconic (perspector X648  = isotomic of Jerabek perspector) and the circumconic with perspector J, the Jerabek center. Since ~a1 and ~a2 are isotomic, the J circumconic is of the Mineur type so that the medials (complements) of these points are on the J circumconic. (Also because of the Mineur designation, the J circumconic has normal lines as asymptotes, in this case the duals (or tripolars) of H and D (= tH).

The medials of the duals
Since ~a1 and ~a2 are isotomic, the J circumconic is of the Mineur type so that the medials (complements) of these points are on the J circumconic. (Also because of the Mineur designation, the J circumconic has normal lines as asymptotes, in this case the duals (or tripolars) of H and D.

The MacBeath inconic
The Jerabek asymptotes go through the major axis vertices X1313 and X1314 of the MacBeath inconic, hence these points lie in the same computational field as the Jerabek asymptotes.

The H and D inconics, and the O circumconic
The axes of these three conics are parallel to the Jerabek asymptotes. Hence they share the same infinite point as the asymptotes.

The Euler intersections with the Circumcircle X1113, X1114 and the Steiner ellipse rX1113, rX1114 and their complements X2454, X2455.

Point notation

a1 and a2 are the Jerabek asymptotes. The D and H inconics have asymptotes parallel to the Jerabek asymptotes. D and H are on the Jerabek hyperbola. The center of the D inconic is O; that of the H inconic is K. O and K are also on the Jerabek hyperbola. We will call o1 and o2 the axes parallel to a1 and a2 through the center O of the D-inconic. Similarly for k1 and k2, through the center K of the H-inconic.

Other lines. We can choose other lines parallel to the Jerabek asymptotes. We will choose ones through H and D, calling them h1, h2, d1, d2, analogous to the others. By choosing points on the Jerabek hyperbola we are using that hyperbola to give a proper spacing to the lines.

Lines

Fissile points (green) are points come in two versions, which together determine a strong line. Fissile lines often concur at a strong point (blue). When the lines are grouped as a pair, they are fissile lines which meet at a strong point. This is the only strong point on either line. If the line is listed singly, it is a strong line on which the fissile points occur in pairs. The name of the line, if any, is at the beginning of the line in red.

dual of Jerabek asymptote endpoint:  G   ~a1   m~a1   ~k2    ~o2    ~h2      ~d2  
dual of Jerabek asymptote endpoint:  G  ~a2     m~a2    ~k1   ~o1    ~h1      ~d1

~J:    S   t879   t(∞•~H)   ~a1    ~a2      tm~a1     tm~a2

m~J:      rJ       mS       m~a1      m~a2

T      1113       ~h1        ~o2
T      1114       ~h2        ~o1

t(∞•~H)         r1113        ~o2
t(∞•~H)        r1114        ~o1

g(∞•~H)     1114      ~o2         ~d2
t(∞•~H)      1113      ~o1          ~d1

g(∞•~J)      r1113      m~a2         tm~a1
t(∞•~J)       r1114       m~a1         tm~a2

t(∞•~H)     r1113      g~h1          ~o1
t(∞•~H)     r1114      g~h2           ~o2

tT      r(∞•e)   tF    ~k1 = t1113       ~k2

J     ~h2       m~a2
J     ~h1       m~a1

S      r1113   1113     ~k1 (= t1113)
S        r1114    1114    ~k2 (= t1114)

tT       r1113     1313   
tT       r1114     1314   

G     tO     t~h2     t~h1   

G     K D     t~o2     t~o1

G     R     t~d2     t~d1  

 

Figure: Master figure for this schema. The Jerabek hyperbola is blue, its switched hyperbola is a lighter blue, the MacBeath inconic (shown as an ellipse) and the dual Jerabek ellipse are both red. The green lines through G are the dual of the infinite points on the Jerabek asymptotes, which are dashed blue.

Table of strong points

The line between two fissile points often contains one strong point. These are given the in table below. The line between the point in the top row and the point in the left column contains the indicated point.
~aJ1
~aJ2
1313
1314
1113
1114
r1113 = r1113
r1114 = r1114
m~a1
m~a2
~k1
=t1113
~k2
=t1114
~o1
~o2
~h1
~h2
~d1
~d2
~aJ1
on ~J
with S,
t(∞•~H),
t879
G
G
G
G
G
~aJ2
G
G
G
G
G
1313
on Euler line
S
tT
T
T
-
1314
tT
S
T
-
T
1113
S
T
T
F
g(∞•~H)
1114
S
T
T
g(∞•~H)
F
r1113
S
tT
t(∞•~J)
S
t(∞•~H)
r1114
tT
S
t(∞•~J)
S
t(∞•~H)
m~a1
G
t(∞•~J)
on rJ—mS line
G
J
G
G
m~a2
G
t(∞•~J)
G
G
J
G
~k1
=t1113
G
S
S
on ~K
the deLongchamps axis with
tT, r(∞•e), tF
~k2
=t1114
G
S
S
~o1
G
T
t(∞•~H)
G
on  ~O
with tF
~o2
G
T
T
t(∞•~H)
G
G
~h1
G
T
-
T
G
J
on ~H
~h2
G
-
T
J
G
~d1
G
F
g(∞•~H)
g(∞•~H)
on ~D
the Polar axis
~d2
G
g(∞•~H)
F
G
g(∞•~H)
G