The Jerabek Hyperbola
and associated structures

What you see in this movie: The B-vertex of the triangle is varied along a circular path. Each time the triangle is isosceles or right the conic degenerates into its asymptotes, and its real foci become the imaginary ones and vice-versa. Also of interest is the motion of the points on the hyperbola itself. notation

Contents Introduction Algebraic Isolation Jerabek hyperbola Properties Organization of Strong Conics Points Incenters of Cevian triangles of Jerabek Pts Points on Rectangular hyperbolas Tangents Conic variables Degeneracies The Mysticum Pascal's Theorem Locus definitions J-circumconic (switched) the dual Jerabek inconic In the field of the Jerabek asymptotes Link to GSP file, begun by Peter Moses with much added by me. This file contains tools for the relevant ETC points, with all versions. Peter Moses contributed much useful information about points on these conics. updated 03/01/07

Figure: The Jerebek hyperbola. The hyperbola is shown in blue. The Steiner ellipse, the circumcircle, the 9 circle, and the auxiliary circle are shown in black. Strong points are shown in blue; 3 fold points such as the triangle and medial triangle vertices in yellow; 2 fold points, such as the foci, in green; and four fold points, such as the Shiffler points H_x (x = o,a,b,c) points in red. Q_1,2 are the intersections of G—K with the Steiner ellipse. pQ_1,2 are their projections to the circumcircle. See point notations below. The dual Jerebek inconic and the switched circumconic (dashed) are shown. The lines D—H, O—H, and ~J, the dual of the Jerebek center, are shown, as well as the hyperbola asymptotes. notation

Introduction
Everyone in triangle geometry knows that the Jerabek hyperbola is important, but is not sure why. Unlike the Kiepert hyperbola, with which it is often mentioned, it is not the result of a dramatic or interesting locus. Rather it is special because of the points it contains, which is rare for a conic. These points organize the associated lines, points, and conics and, in my opinion, explain much of what goes on in the triangle plane. This paper covers this and more.

We generate circumconics as the isotomic conjugate of a line. This relationship between the generating line and he generated conic is affinely invariant.

A special type of circumconic, which I call a Mineur conic, is obtained by connecting a line to its conjugate (the Mineur line) then taking the conjugate of that line. The conjugate points that define the line lie on the conic and define the associated affine parallelogram. A Mineur conic contains a highly symmetric group of 4 points that define and organize related circumconics and inconics. In this case the related conics are the strong ones.

The Jerabek hyperbola organizes the strong conics, which include those associated with the names Kiepert, MacBeath, Brocard, as well as the circumcircle and the H and D inconics.

Notation can be found here. The generic term "conjugate" will refer to the isotomic conjugate.

Algebraic considerations
Center, perspector, axes, asymptotes, foci, directrix.

The triangle vertex coordinates exist in a certain computational field, often the rationals (represented by Q in the Figure below) since we tend to choose A, B, C to have integer coordinates. But computations often require the field to be extended. For example triangle edgelengths are square roots, so that coordinates that use edgelengths in their definition must exist in a different and larger field than the triangle vertices, the rationals extended by the 3 distances. Often the perspector and center of a conic exist in this field.

The asymptotes and axes of the conic require knowledge of the intersection of a line with the conic, in both cases requiring a (different) larger field. The vertices are now intersections of the axes with the circumconic, requiring yet another field extension.

The figure below shows the field extensions that usually have to be done to create various conic objects.

We should note that here we are constructing a conic from its perspector. Other starting choices are possible, for example one could begin with the foci.

The affects of this algebraic structure are profound. Points and lines from fields on the right of the following chart barely interact with those on the left. If one has configuration with many known triangle points and lines (generally constructed in a simpler fields) with many asymptotes and foci (in more complicated fields), the foci will not be on the lines and the points not on any of the asymptotes. These objects are algebraically isolated from each other.

Algebraic isolation

The farther along the chain of field extensions an object is, the more isolated it is from objects that live in earlier fields. Hence the foci, for example, seldom lie on famous lines or famous points on conic axes.

Figure: Field extensions for an arbitrary circumconic. Each arrow represents a field extension, given in red. The farther along the chain of arrows, the more algebraically isolated the object!

Perspector and Center: these exist in an algebraically harmonious world, the center existing in the same field as the perspector, usually the field of the vertices extended by the three edgelengths.

Asymptotes and axes, which require a root field extension incommensurate with other points, are constructed here from the intersection of a line with the Steiner ellipse.

Vertices, Foci, and directrices. These objects require another square roots and hence more field extensions. These points and lines are truly isolated from the others. It is very difficult for foci or directrices, to relate to much of anything except points generated by their own field extensions. (For a situation with lots of points related to foci, see here).

Properties of the Jerabek hyperbola
and some other inconics, circumconics, and inparabolae.

1. The Jerabek hyperbola is the Mineur conic from the line from H to its conjugate D = tH. Since its generating line goes through D, it is a rectangular hyperbola.

It is a Mineur conic in a second sense because it is the isogonal conjugate of the Euler line, which joints the isogonal conjugates O and H.

Its perspector is the dual of the D—H line: JP = (: b²(c²–a²) S_B :)

a²(b²–c²) S_A / x + b²(c²–a²) S_B / y + c²(a²–b²) S_C / z = 0

Note: the Jerabek perspector with b² replaced by b, a² with a, and c² with c becomes the Feuerbach perspector. This preserves its affine form so many of the properties of these conics are similar. We say the Feuerbach hyperbola is the weakened form of the Jerabek one.

2. Special points: As the conjugate of a line joining the conjugate points H, D, the points H, D, mH = O, and mD = K are on the conic. This is a unique set of four points as explained below.

As the isogonal conjugate of a line joining a point to its isogonal conjugate, O—H in this case, the points O, H, suppO = X65 (= gH_o, the isogonal of the Schiffler point), and supp H = X73, where supp is the supplementaire operation, the operation that goes with the isogonal conjugate in the same way that medial goes with the isotomic conjugate.

The path of the Jerabek hyperbola follows the affine parallelogram for H. It is also related to the path through the triangle I call the sweep of the incenter. The four points H, O, K, D are both shared by the affine parallelogram of H and the Jerabek hyperbola. The Mineur line H—D is one diagonal, while the other diagonal goes through G, F, and J = mF, the hyperbola center.

The four points H, D, O, K on the hyperbola define most of the important strong circumconics and inconics. This is an example of the way a Mineur conic defines conics relevant to particular points.

Figure: This picture shows how the Jerabek hyperbola goes through points of the H-affine parallelogram. The Jerabek center lies on the second diagonal of the parallelogram.

3. The perspector. Tripolars of the points on the Jerabek hyperbola go through the Jerabek perspector JP. In particular the tripolars of H, O, K, D go through this point. The duals of points on D—H go through this point. As is true for all rectangular hyperbolae, the perspector is on the polar axis (tripolar of H = ~D). It is also on the Lemoine latitude line (tripolar of K).

The duals of H and D go through the Jerabek perspector. The duals of K and O go through tF, where F = pS is the focus of the Kiepert parabola.

The edges of the preCevian triangle of the perspector are the tangents to the hyperbola at A, B, C.

4 Lines between the Jerabek special points and special points on the Steiner ellipse and circumcircle.
A point P is naturally associated with two lines, its dual ~P, and G—P. The conjugates of the infinite points (directions) of these lines being antipodal on the Steiner ellipse.

For the Jerabek hyperbola H and D = tH eachcreate two pairsof antipodal points on the Steiner ellipse. These points and the special points on the hyperbola give a nice collection of interrelated lines and points.

Figure: the duals (also tripolars) of P and tP meet at the Mineur point, which is the perspector of the Mineur conic, generated by the P—tP, the Mineur line. Notation.

For the Jerabek conic, the isotomic points are D and H. The four Jerabek special points K, D, H, and O turn out to be on the lines connecting the points on the Steiner ellipse, and a corresponding set of points on the circumcircle, as seen in the figure.

Figure: Lines and points. Antipodal pairs of points on the Steiner ellipse related related to the points D and H. Lines through these points go through G and the four special Jerabek points.
-H and -D are the reflections in the conic center of H and D.
k and e are the G—K line (the symmedian track) and the Euler line. Solid black lines are between points on the Steiner ellipse. Dashed black lines are connections between points on the circumcircle. The red dashed lines are the Jerabek asymptotes. The relation of points and lines happens for all conics of the Mineur type, but is particularly powerful for the Jerabek example because it is a Mineur conic defined by both the isotomic pair H, D and the isogonal pair O, H. Notation.

5. The center J = mtd JP = (: (c²–a²)²S_B :). The center can be constructed as the medial of the intersection of the second diagonal of the affine parallelogram with the circumcircle. As a rectangular hyperbola the center is on the 9-point circle.

The dual of J goes through t(∞•~H) and t(∞•~D), which are the conjugates of the directions of the duals ~D and ~H and are on the Steiner ellipse.
~J is perpendicular to T—H.

6. The asymptotes a₁ and a₂ are the Simpson lines of the meets of O—H with the circumcircle. Their directions are the conjugates of the meets of the D—H line with the Steiner ellipse

The equations of the asymptotes are

[ : b²(c² – a²) S_B (c² a² (–a² b⁶ + b⁸ + a⁶c² + a²b⁴c² – b⁶c² – 2a⁴c⁴ + a²c⁶) ± b²(a⁴ – b²c²+c⁴–a²b²) Z)² :]
where Z = √(a²b²c² (S²S_W–9S_ABC)).

6.5 The directions of the Jerabek asymptotes. For a rectangular Mineur hyperbola, such as the Jerabek, the directions of the asymptotes are also the directions of the axes of the circumconics of the inner points O and K and the inconics of the outer points D and H. This is an impressive property, again showing the organizing property of a Mineur conic.

This pleathora of lines parallel to the Jerabek asymptotes will have intersections and duals which will also be in the field of these asymptotes. There will be many interesting connections between between these points and lines. I have written an article on this.

7. The Jerabek hyperbola has axes parallel to the asymptotes of the rectangular hyperbola with perspector ( : b²(2b⁴S_B – 4S_ABC – b² S4_B) : ), where S4_B = (a⁴–b⁴+c⁴)/2. This point is on the polar axis. Since it is rectangular the axes are the angle bisectors of the asymptotes. The Jerabek asymptotes are also parallel to the axes of the H and D inconics and the O circumconics. These points are on the hyperbola.

8. The Jerabek hyperbola is tangent to the Lucas cubic at H and the McCay cubic at ??. This follows from the relation of Mineur conics to pivotal cubics, which is that the Mineur conic is tangent to the pivotal cubic whose pivot is the dual of the Mineur line; i.e., the perspector of the conic. Remember the Jerabek hyperbola is a Mineur conic in two ways and so is the tangent conic for two cubics.

A point on the Lucas cubic is collinear with D (the pivot) and the conjugate of the point. Each line thus generates a rectangular hyperbola of the Mineur type. One can make much of this.

9. The dual inconic has perspector tJP and center mJP. Its center is on the D—S line and the second diagonal of the affine parallelogram of D and H. The dual of any point on the Jerabek Hyperbola is tangent to this conic. In particular this applies to the duals of D, H, O, K.

The best known points on this (shown in the table below) are the points of tangency of the duals of the four main points on the Jerabek hyperbola: ~K (tX112), ~D (X2489), ~O , ~H

10. The switched circumconic has perspector J and center JP. It is generated by the line ~J, the dual of the hyperbola center, which connects the duals of the Jerabek asymptotes ~a₁ and ~a₂, and is parallel to G—K and perpendicular to T—H. This is a Mineur conic generated by ~J. m(~a₁) and m(~a₁) are thus on the switched conic. The asymptotes of the switched conic are ~D, the Polar axis, and ~H, two well known lines.

If the isotomic points of a Mineur hyperbola are known, rational points, then the asymptotes of its switched conic will be known, rational lines. This means that one can find and state all hypebolas with well known lines as asymptotes.

Points on J-circumconic [from Peter Moses] center 647, perspector 125
{523 tS, 525 ∞•~H, 879 (b²–c²) S_A / (b^4– + c^4–) , 935, 2394 (b²–c²) / (a² S_A–2S_BC), 2966}

11. The last two conics meet at four points, two on the dual of J which are the conjugate points ~a₁ and ~a_2, the duals of the Jerabek hyperbola asymptotes. The other two are the points of tangency of ~O and ~K with the inconic.

The lines from G to the isotomic points ~a₁ and ~a₂ are tangent to the dual inconic.

12. The dual of the center ~J goes through t(∞•~H) and S = t(∞•~D) and the asymptote duals mentioned in the last item.

13. Two associated inparabolae:
The lines G—JP and G—J each have an associated circumhyperbola and dual inparabola. The two inparabolae are particularly interesting:

From G—JP:
inparabola with perspector = t(∞•(D—H)) = X265 , the intersection of the Jerabek hyperbola and the Steiner ellipse.
Focus the Tarry point T.
Axis through T and K^2– parallel to D—H.
Directrix through H and the 4th intersection of Jerabek hyperbola and its switched circumconic (: (c²–a²) S_B / (c⁴+a⁴–c²b²–a²b²):)

From G—J:
inparabola with perspector t(∞•~J), the intersection of the switched conic and the Steiner ellipse: (: 1/(c²–a²) (c⁴+a⁴–c²b²–a²b²):) ;
Focus, the pro of this point which is on the circumcircle;
Axis parallel to ~J through the Focus;
directrix, the line H—T, through mS and pK.

Figure: inparabolae. Two inscribed parabolas are shown in red near vertex B. The Jerabek dual is shown in red at the bottom. Notation.

14. Conics of points on the Jerabek hyperbola

The inconics of D and H go through J.
The D-inconic has axes parallel to the Jerabek asymptotes (I learned this from Bernard Gibert).
The centers of inconics of O and K are collinear. with O and K.

The circumconics of O and K meet on the G—J line, the second axis of the D, H affine parallelogram, at the point F = dJ on the circumcircle.
The circumconic of O goes through the reflection in J of g(∞•e) = t(∞•~H) and the reflection of D, which is on the Jerabek hyperbola. The axes of the O-circumconic, center K, are parallel to the Jerabek asymptotes.
The circumconics of D and H meet at tJP where they are tangent to ~J.

15. The tangents at the vertices of a circumconic are the ex-Cevian lines of the perspector. The tangents at the intersections of G—J with the Jerabek hyperbola are parallel to D—H.

16. 4th intersections: The fourth intersection of Jerabek hyperbola with the circumcircle g(∞•e) (X74) is the isogonal of the Mineur endpoint. The Steiner intersection is the isotomic of the Mineur endpoint t(∞•(D—H)) = X265.

17. The parallelogram: O, H, g(∞•e) (X74) , and gvH = X265 form a parallelogram with sides parallel to the Euler line and to the line F–O–g(∞•e) where F is the Kiepert parabola focus and e is the Euler line.

18. Strong and SuperStrong. The points H and its strengthened version sH = (: 1/(c⁴ + a⁴ – b⁴) :) are both on the Jerabek hyperbola. We sometimes use the shorthand S4_B = (c⁴ + a⁴ – b⁴)/2. This has the effect the Feuerbach hyperbola, the weakened Jerabek hyperbola, also contains H and is rectangular.

Master Figure: The red line is the isotomic Mineur line D—H; the blue line is the isogonal Mineur line O—H. The blue hyperbola is the Jerabek. The red inconic is its dual. The green parallelogram is the affine parallelogram of D and H. Q, R define the second diagonal on which J, the hyperbola center, and F, the focus of the Kiepert parabola lie. The blue parallelogram connects 4 special points. Notation.

Organizing the strong conics
points on the Jerabek Hyperbola organize the strong conics.

A Mineur conic, such as the Jerabek hyperbola, organizes other conics associated with its most prominent points. A Mineur conic is defined as the conjugate of a line from P to its conjugate tP. mP and mtP are also on the conic. These four points carry an exceptional amount of structure with them and determine the a large number of conics relevant to the point P. For the Jerabek hyperbola, we let P = K the symmedian point.

The four points are on, and define, the affine parallelogram for the conjugate pair.

Using the above operations, consider K,
and then its dilated (or antimedial): dK = D,
and then the conjugate of this: H,
and finally the medial: O.

4 points; six pairs of points.
One pair (D, H) is conjugate, defining the Mineur line that generates the Jerabek Hyperbola, which contains the four points.
Two pairs (K, D and H, O) have the medial/dilated relation and are collinear. with G. These lines define conics through G with perspectors at infinity, one being the Kiepert hyperbola.
Three pairs (K, O and K, H and O, D) have the perspector/center relation, one for a pair of circumconics (the circumcircle and O circumconic) and two for inconics (the D and H inconics).

One pair are isogonal conjugates, being the foci of an inconic. The MacBeath inconic has foci O and H.

I do not think it is possible for four points to carry more structure.

Figure: The web of relationships relating the strong conics to the four principle points on the Jerabek hyperbola. The lines OGH and KGD define prominent hyperbolas and inparabolae.

Figure: the axes (red, dashed) of the O and K circumconics as well as the H and D inconics are parallel to the Jerabek asymptotes. Notation. The axes meet on the Brocard circle.

More Points on the Jerabek hyperbola, the dual Jerabek inconic, and the switched Jerabek circumconic

Peter Moses lists points on the Jerabek hyperbola as ( I have labeled strong ones blue, weak ones red, and the one super-strong point is black, and greyed out points that I have not thought important enough to include): (Notation.)

Jerabek circumhyperbola, center 125, perspector 647
{3,4,6, 54 (gN a² /(a²S_A+2S_BC)) , 64 (gL) , 65 (gH_o), 66 (sH), 67 (gvG), 68 (g24), 69 D, 70 ,71,72,73 (g29), 74 (g e∞),248, a²S_A/(b^4–+ c^4–) , 265 (gvH) S_A/(4S_A²-b²c²),290 rT, 695 (gK²⁺), 879 (b²-c²) S_A/(b^4–+c^4–) , 895 D' ={a²S_A/(b²+c²–2a²)}, 1173 {a² / (2 S_BC + 3 a² S_A)}, 1175, 1176 (b²S_B/(c²+a²) = sH_o), 1177,1242,1243,1244,1245,1246, 1439 {a (b+c) S_A / s_a²}, 1798 {a² S_A / (b+c)(b²⁺+c²⁺)}, 1903 { a (b+c)/(–2_sabc+a S_A)}, 1942, 1987, 2213, 2435, 2574,2575,2992,2993}

Notes: X(54) =  gN
X(64)  = gL = b²/(b²S_B–S_CA)
X(71) = ISOGONAL CONJUGATE OF X(27) = b² (c + a) S_B
X(72) = ISOGONAL CONJUGATE OF X(28) = b(c + a) S_B
X(895) = ISOGONAL CONJUGATE OF X(468) = b²(c²+a²–b²)/(c²+a²–2b²).

Here is a link to my value added version of Peter's compilations.

125 circumconic (Jerabek switched), center 647, perspector 125
{523 tS, 525 ∞•~H, 879 (b²-c²)S_A/(b^4–+ c^4–) , 935 1/S_A(b²-c²) (2 S4_A-b²c²), 2394 (b²-c²)/(a²S_A-2S_BC), 2966 1/(b²-c²)(b^4–+c^4–)}

The points on the dual Jerabek conic are probably not relevant to much known geometry. I include them because I think that the dual conic is more interesting than I thought it would be.

The following chart organizes both points and lines. Each entry has the y coordinate (in black) of some line or point. The ETC number is red and the Geometry interpretation, if known, below. Each entry can be doubly interpreted, as a point for some conic or as a line for the dual conic. The top row in black and yellow indicates the conic for which the entry represents a point; the second row in dual colors yellow and black indicates the conic for which the entry represents a tangent line. For example ( : b²(c+a)S_B : ) is the y coordinate of a point on the Jerabek hyperbola and [ : b²(c+a)S_B : ] is the y-coordinate of the line tangent to the dual Jerabek inconic.
The colors divide the chart into sections whose structure can be said to originate from the named points, as explained here.

Notation.

line at infinity	point on Jerabek Hyperbola	point on dual Jerabek	J-Hyperbola switched Jerabek
	tangent to dual Jerabek	tangent to Jerabek Hyperbola
(:m:)	: b2(c2–a2)SB/m :	: m2 / b2(c2–a2)SB :	(: (c2–a2)2SB/m :)
		from incenter
514 c–a ∞•~Io	71 b2(c+a)SB	? (c–a)/b2(c+a)SB -	? (c+a)(c2–a2)SB 879-71
513 b(c–a) twS = ∞•~tIo	72 b SB/(c+a) -	? b2(c–a)/b2(c+a)SB -	? (c2–a2)SB/b(c+a) 879-72
900 (c-a)(c+a-2b) (∞•~190o)	? b2(c+a)SB/(c+a–2b) 4, 1320	? (c-a)(c+a-2b)2/b2(c+a))SB -	? (c+a)(c2–a2)SB/(c+a–2b) -
812 (c–a)(b2–ca)	? b2(c+a)SB/(b2–ca) 4, 294	? (c–a)(b2–ca)2/b2(c+a)SB	? (c+a)(c2–a2)SB/(b2–ca) -
? (c+a)(b2–ca) ∞•(Io—tIo)	? b2(c–a)SB/(b2–ca)	? (c+a)(b2–ca)2/b2(c–a)SB -	? (c–a)(c2–a2)SB/(b2–ca)
? b(c-a)sbb, ∞•~Go2	1439 b(c+a)SB/sbb, -	? b2(c-a)sbbbb/b2(c+a)SB, -	- (c+a)(c2–a2)SB/bsbb, 879-1439
521 b(c–a)sbSB	65 b(c+a)/sb HGo, orthocenter of intouch ∆	? b2(c–a)sbbSB/b2(c+a)	- (c+a)(c2–a2)/bsb 879-65
		from Gergonne Point
522 (c–a)sb ∞•~Go	73 b2(c+a)SB/sb	? (c–a)sbb/b2(c+a)SB -	? (c+a)(c2–a2)SB/sb 879-73
		from Symmedian Point
523 c2–a2 tS = ∞•~K	3 b2SB O	: (c2–a2)/b2SB:	525 (c2–a2)SB ∞•~H
512 b2(c2-a2) gS = ∞•~tK	69 SB D = tH = dK	2489 : b2(c2–a2)/SB:	935 (c2–a2)SB/b2 4th CC
(c2-a2)(c2+a2-2b2)	895 b2SB/(c2+a2–2b2) reflection of D in J.		(c2–a2)SB/(c2+a2-2b2)
? (c2–a2)(c4–+a4–) -	248 b2 SB/(c4–+a4–) -	? :(c2–a2)(c4–+a4–)2/b2SB: -	879 (c2–a2)SB/(c4–+a4–) 4th JH
? (c2–a2) b4– -	? b2 SB/b4– -	? :(c2–a2)( b4–)2/b2SB: -	? (c2–a2) SB/b4– -
(c2–a2)3SB(c4–+a4–)	b2/(c2–a2)2(c4–+ a4–)		2966 1/(b2–c2)(b4–+ c4–) t(∞•~J) 4th SE
? b4(c2–a2)SB(c4–+a4–) -	290 1/b2(c4–+a4–) rT		? (c2–a2)/b4(c4–+a4–) -
gT	879 (c2–a2)SB/(c4–+ a4–)		? (c2-a2)2SB/b2 (c4–+a4–)
(c2–a2)SB b4+	879 b2/b4+ gK2+		? (c2-a2)2SB/b2 b4+
s521 b2(c2–a2)SBS4B	66 1/S4B sH
		from Orthocenter
525 (c2–a2)SB ∞•~H = ∞•~O	6 b2 K	t112 (c2–a2)SB/b2 -	523 (c2–a2) ∞•~D = ∞•~K
30 SBC+SAB–2SCA ∞•(G—H) infinite point on Euler line	b2/(SBC+SAB–2SCA)	? (SBC+SAB–2SCA)2 / b2(c2–a2)SB -	2394 (c2–a2)SB/(SBC+SAB–2SCA)
-(c2–a2)SB(b2 SB–m SCA) point on Euler line	b2/(b2 SB–m SCA) includes 54 (gN), 64 (gL) and others	? (b2 SB–m SCA)2 / b2(c2–a2)SB -	? (c2–a2)SB/(b2 SB–m SCA)
520 b2(c2–a2)SBB ∞•~H2	4 1/SB H	? b2(c2–a2)SBBB -	? (c2–a2)/b2SB -
		from pK
? c4–a4 tsS = ∞•~pK	1176 b2SB/(c2+a2) sHo
b2(c2–a2)SB(2S4B–b2c2)	67 1/(2 S4B–c2a2) gvG

The incenters of Cevian triangles of points on the Jerabek hyperbola

Hyacinthos #7047 (Jean-Pierre Ehrman)
The in/excentres of the cevian triangle of M lie on the rectangular circumhyperbola through M

[Bernard Gibert] It follows that the points which have an in/excenter of their cevian triangle at K must lie on the Jerabek hyperbola. This can be generalized for any other point such as O and D. I doubt very much these points can be constructed with ruler and compass.

Proof [Bernard Gibert]: The rectangular hyperbola passing through P and the 4 in/excenters of P_aP_bP_c is a diagonal conic wrt P_aP_bP_c. It must contain the vertices of the anticevian triangle (wrt P_aP_bP_c) of any of its points and in particular it contains A, B, C.

Figure: A_K is a vertex of the K-Cevian triangle. I_aK is one of the 4 incenters of this triangle, which is on the Jerabek hyperbola.

Correspondence of points on Kiepert, Jerabek, and Feuerbach hyperbolas.

The line between corresponding points on two circumconics goes through their 4th intersection. Here is a picture of the Kiepert, Jerabek, and Feuerbach hyperbolas, all of which have a fourth intersection at H, with corresponding points joined by lines through H. Kiepert is blue, Jerabek red, and Feuerbach green.

Figure: The red hyperbola is the Kiepert one, green the Feuerbach, and red the Jerabek. Note that the corresponding points from the above table are on lines through H.

Figure: Here the red hyperbola is Kiepert, purple Jerabek. The four blue ones are the Feuerbach hyperbolas. Red points are weak, blue strong, green fissile, and cyan octile. The dashed lines are through H and connect corresponding points on the six hyperbolas. Notation.

The points in the following table is indexed by ETC number in red, which is included when I have the energy to find it and, if available, an indication of how to construct the point. Numbers like 1,7 indicate a line the point is on. If possible, the construction is the second intersection of this line with the conic. The second barycentric coordinate is given in black.

By my theory of infinite points, points on conics belong to families that can be associated with triangle centers. Usually only a few famous centers suffice to generate a reasonable set of points on the conic.

A special note about points on rectangular hyperbolas

The Kiepert hyperbola, the Jerabek hyperbola, and the Feuerbach hyperbolas are rectangular hyperbolas, intersecting at H, which is their fourth intersection, each with the other. The fourth intersection point is the fixed point of the projective mapping that takes one conic to the other. It can be implemented as a line between the point and it image, which always goes through H.

A version of this chart is reproduced on the Kiepert and Jerabek pages, optimized for the particular conic.

Notation.

line at infinity	Kiepert Hyperbola	Feuerbach Hyperbola	Jerabek Hyperbola
(:m:)	(: (c2-a2)/m :)	(: b(c–a)sb/m :)	(: b2(c2–a2)SB/m :)
		From the Incenter
514 c–a ∞•~Io	10 c+a Spieker point So	9 b sb Mo the Mittenpunkt	71 b2(c+a)SB
513 b(c–a) twS = ∞•~tIo	321 (c+a)/b -	8 sb No	72 b SB/(c+a) -
900 (c-a)(c+a-2b) (∞•~190o)	? (c+a)/(c+a-2b) -	1320 b sb/(c+a–2b)	? b2(c+a)SB/(c+a–2b) 4, 1320
812 (c–a)(b2–ca)	812 (c+a)/(b2–ca)	294 b sb/(b2–ca) 6,7	? b2(c+a)SB/(b2–ca) 4, 294
? (c+a)(b2–ca) ∞•(Io—tIo)	? (c–a)/(b2–ca) -	? b/(c+a)(b2–ca)	? b2(c–a)SB/(b2–ca)
? b(c-a)sbb, ∞•~Go2	1446 (c+a)/bsbb,	7 1/sb, Go	1439 b(c+a)SB/sbb, -
521 b(c–a)sbSB	? (c+a)/bsbSB	4 1/SB H	65 b(c+a)/sb HGo, orthocenter of intouch ∆
? (c–a) sb (b2+bc+ab+2ca)	? (c+a)/(b2+bc+ab+2ca)	941 b/(b2+bc+ab+2ca) 6, 21	? b2(c+a)SB/(b2+bc+ab+2ca) 4, 941
? b2(c–a)sb(c2+a2–ab–bc)		2481 1/b(c2+a2–ab–bc) 4th intersection with SE	? (c+a)SB/sb(c2+a2–ab–bc) -
		From the Gergonne point also the Mittenpunkt
522 (c–a)sb ∞•~Go	226 (c+a)/sb -	1 b	73 b2(c+a)SB/sb
? (c–a) sb (c2+a2–bc–ab–2ca) =(c–a) sb (csc+asa)		2346 b/(csc+asa) 7, 55
		From the Symmedian point
523 c2–a2 tS = ∞•~K	2 1 G	21 b sb/(c+a) Ho	3 b2SB O
512 b2(c2-a2) gS = ∞•~tK	76 1/b2 R = tK	314 sb/b(c+a) 6, 98 collinear	69 SB D = tH = dK
? (c2–a2)(c4+a4–a2b2–b2c2) -	98 1/(c4+a4–a2b2–b2c2) T	? b sb/(c+a)(c4+a4–a2b2–b2c2) -	248 b2 SB/(c4+a4–a2b2–b2c2) -
? (c2–a2)(b4–c2a2) -	1916 1/(b4–c2a2) tK2–	? bsb/(c+a)(b4–c2a2) -	? b2 SB/(b4–c2a2) -
		From the Orthocenter also the Circumcenter
525 (c2–a2)SB ∞•~H = ∞•~O	4 1/SB H	1172 b sb/(c+a)SB 4, 6 collinear	6 b2 K
30 SBC+SAB–2SCA ∞•(G—K) infinite point on Euler line	2394 (c2–a2)/(SBC+SAB–2SCA)	b(c–a)sb/(SBC+SAB–2SCA)	b2 SB/(SBC+SAB–2SCA)
520 b2(c2–a2)SBB ∞•~H2	2052 1/b2SBB rH2	1896 sb/b(c+a)SBB	4 1/SB H
		From pK
? c4–a4 tsS = ∞•~pK	83 1/(c2+a2) tmK		1176 b2SB/(c2+a2) sHo
? b4(c4–a4) ∞•~tpK	? 1/b4(c2+a2) -		? SB/b2(c2+a2) -
? (c2–a2)(c4-c2 a2+a4-b4) -	? 1/(c4-c2 a2+a4-b4) gvG		? b2SB/(c4-c2 a2+a4-b4) -
		Fissile points
		1251 b/(sca+√3 ∆) 1, 15 octile
	Fn
		177 (octile)

MacBeath in relation to the Jerabek hyperbola.

The Jerabek hyperbola contains the points H, K, O, D of which H, O are isogonally conjugate. H and O are the two foci for the MacBeath inconic.

As we would expect there are many connections between these conics.

1. The asymptotes of Jerabek go through X1312 and X1313, the vertices of MacBeath. This means that vertices are in the same algebraic field as the Jerabek asymptotes.
2. The axes of the H and D inconics are parallel to those of Jerabek.
3. The orthic (H) inconic meets MacBeath at the four versions of X2969.
4. The D-inconic meets MacBeath at the four versions of X2968.
5. The H and D inconics go through the Jerabek center.
6. The MacBeath circumconic (center K, perspector O) has axes parallel to the Jerabek asymptotes.

Figure: The Jerabek hyperbola is blue. The MacBeath inconic is red. The D and H inconics are black. Weak points are red, strong, blue, and fissile green. Note that the asymptotes of the hyperbola go through 1312, 1313, the major vertices of the MacBeath ellipse.

Tangents

The tangents to the Jerabek hyperbola at A, B, C form the ex-Cevian triangle of the Jerabek perspector. A nice way to construct them is the following. The lines A_H—C_K and C_H—A_K meet on the intersection of K—H line and the B tangent, which gives a construction for that tangent. Note: B_H is Conway notation for the B-trace of the orthocenter.

This constuction is works for any two points on the hyperbola.

A_H B_K C_H A_K B_H C_K are con-conic and form a Third hexagon which has 3 double intersections with the K—H line at the intersections of this line with the Jerabek asymptotes. This new conic, which we will call the invariant conic, will be discussed in the next section along with this special parallelogram for which the H—K line is the Pascal line.

Notation.

Figure: The green lines are edges of the hexagon AAKHBC. The intersection of opposite sides define the Pascal line in red. The Pascal line is indexed by its hexagon. Notation.

This picture is shows 3 Pascal lines, each labeled by the six vertices of its hexagon, converging on a Steiner point, labeled by the three unchanged vertices in the three Pascal lines. The technique for doing this is explained here.

Figure: Pascal lines converging on a Steiner point. The letters next to each line give the hexagon that defines and indexes the Pascal line. AKB, the unchanged elements of all three indices, is notation for the Steiner point, explained here.

The next picture below show how the Jerabek Hyperbola organizes miscellaneous points via Pascal's theorem. This picture shows that whole Hexagrammum Mysticum for the AACKHB hexagon. The colored points and lines represent the color symmetry of the Kirkman points and Pascal lines.

 

The table, adapted from Jean Peyroux, shows 5 points on each tangent lines, any one of which could be used to construct the line.

Here h_a = altitude through A.
m_a = median through A.
k_a = symmedian through A.
k = HK line.
a = A-edge.
A_G = A-trace of G = midpoint on A-edge.
A_H = A-trace of H = foot of A-altitude.
A_K = A-trace of K.

Hexagrammum mysticum centers from the invariant Cevian conics
central lines and central points

Pascal's theorem posits a line for each hexagram created from six points on a conic. Since there are 60 non-equivalent hexagons, there are 60 distinct lines, which concur at 20 Steiner points and 60 Kirkman points, which determine 15 Plucker lines and 20 Cayley lines, which determine 15 Salmon points. The collective structure of 190 points and lines is the Hexagrammum Mysticum.

The members of the mysticum are never central points or lines for a circumconic (as shown below). But each circumconic determines another conic which does produce central points.

A circumconic is completely determined by 5 points, ABC and two others which we will call P and Q. The Cevian triangle vertices of P and Q form 6 points that are on another conic. For the Jerabek hyperbola ABCKH, this Cevian conic goes through the points A_H B_K C_H A_K B_H C_K. John Conway has pointed out that the hexagon from these points, taken in this order, form a very special hexagon which is invariant under permutations of ABC, which means that the Pascal line that it creates is a central line. The associated Steiner point, Cayley line, Plucker point, Kirkman point, and Salmon point are also central. John calls these objects thePascal, theSteiner, thePlucker, etc. There is a second hexagon with this property, giving a second set of central objects. This set is named the coPascal, coSteiner, coPlucker, etc.

For the Jerabek ABCKDHO hyperbola, computation gives these points and lines, taking all combinations of the K,D,H,O for P and Q. These are shown in the following chart of charts. The K,O and D, O tables are the most interesting. Entries not filled in have complicated formulas.

Notation.

	K	D	H	O
K	–	the co Pascal G–K Steiner Kirkman Plucker	the co Pascal [:(c2–a2)SB2:] K—H Steiner Kirkman Plucker	the co Pascal [ : (c2–a2)/b2: ] OK A-median Steiner gT at infinity centroid Kirkman O (:(c2–a2)SB2–:) Plucker [:b2(c2–a2):] Lemoine line ∞
D		–	the co Pascal [:b2(c2–a2)SB:] D—H Steiner Kirkman Cayley Plucker	the co Pascal [:(c2–a2)/SB:] D—O Steiner (:(b4–c2a2)SB:) (:SB/(c2+a2):) Kirkman (:b2(c2+a2)SB:) (:b2(c2–a2)(b4–)SB:) Plucker [:(c2+a2)/SB:]
H			–	the co Pascal Euler line Steiner Kirkman Plucker
O				––
	K	D	H	O

Jerabek conic variables and degenerate behavior

Peter Moses has computed the radius of the auxiliary circle for the Jerabek hyperbola which is

[PM]
T = (a² – b²) (b² – c²) (c² – a²)
S = twice area ABC,
The squared radius of the "incircle" of Jerabek is
T S_ABC / (4 a b c OH³ S²)
OH = distance circumcenter to orthocenter.

Here are the conic variables for this hyperbola Notation.

The Jerabek hyperbola
perspector	JP = ( :b2(c2 – a2)SB: )
center	J = mtd JP = ( : b2(c2 – a2)2 SB:)
semi-major axis	√(T SABC / (4 a b c OH3 S2))
semi-minor axis	√(T SABC / (4 a b c OH3 S2))
semi-focal	√(T SABC / (2 a b c OH3 S2))
radius of director circle	0

We see from the T factor that the radius of the auxiliary circle, goes to zero as the triangle becomes isosceles. The hyperbola degerates, becoming a median and an edge. We see from the S_ABC factor the radius goes to zero if the triangle is right. The hyperbola again degerates, becoming an altitude and an edge. Here is a movie showing this behavior. This movie (2M) is formatted for the iPod. A larger one (30 M) is here.

Locus Definition

From Casey: Let ∆^K be the tangential triangle, whose edges are tangents to the cirucmcircle at the vertices. The notation indicates that this triangle is known as the preCevian triangle of K, the symmedian point. The locus of the perspectors of homothetic triangles to ∆^K is a rectangular hyperbola known as the Jerabek Hyperbola.

It is the isogonal conjugate of the Euler line.

The locus of points whose tripolars go through the Jerabek perspector is the Jerabek hyperbola.

Anyway, here is a locus :
L_a, L_b, L_c are // to orthic triangle's sidelines,
concurrent at P.
Let L_a /\ AB = A_b, and L_a /\ AC = A_c
Similarly B_c, B_a; C_a, C_b.
(if the points are concyclic we have a Tucker hexagon)
What is the locus of P such that the triangles AbBcCa
and A_cB_aC_b have equal areas?
The answer is the Jerabek hyperbola.

Impressed?