The Feuerbach hyperbolas
and associated conics

Contents Introduction Properties Points The Feuerbach organizes the weak conics. Points on rectangular hyperbolas Desmic relations Cubics Hexagrammum Mysticum and the Invariant Conics Locus Links:

(updated 2/20/07) The intent here is to investigate a weak conic in all four of its versions. To my knowledge this has never been done, and represents a glaring weakness in our knowledge of these objects. I think we do not investigate all versions since we take the central region of the triangle too seriously. This has to be a successful investigation since there will automatically be correlations in the points involved. Multiple versions of objects are covered here.

I chose the Feuerbach circumhyperbola since, other than the incircle, it is the best known weak conic. It goes through H, and hence is rectangular. It is known that the perspector of a regular circumhyperbola is on the Polar axis and its center is on the nine point circle. It is also a Mineur conic, a conic of high symmetry generated as the isotomic conjugate of the Nagel-Gergonne line. The general theory of rectangular and Mineur conics is here.

Notation is here.

The left picture is the original Feuerbach hyperbola with the four best known points on it: the incenter, Gergonne, Nagel and Mittenpunkt points. It's path through the center of the triangle is what I call the sweep of the incenter. The picture on the right shows all four Feuerbach hyperbolas, and shows the 4 centers on the nine point circle and the perspectors (P_x where x = o,a,b,c) on the polar axis. The picture below shows all four hyperbolas for an obtuse triangle, which clearly shows that the perspectors lie on the Polar axis, the radical axis of the circumcircle and the 9 point circle. The four hyperbolas are tangent at H.

This figure clearly shows that the polar axis (the line through the intersection of the nine point circle and the circumcircle, is the line of the perspectors, one of which is off-picture.

The 4 Feuerbach hyperbolas

We begin with a nice result.

1. The 4 perspectors are collinear.

These four conics are generated as the isotomic conjugate of the lines from a Gergonne point G_x to its isotomic conjugate, the corresponding Nagel point N_x, where x = o,a,b,c indexes the four versions. A line from a point to its conjugate possesses high symmetry. I call it the Mineur line after work by Adolph Mineur found here. The perspectors are the duals of the Mineur lines m_x = G_x—N_x, which are P_o = (: b(c-a)s_b :) and extraversions. The 4 Mineur lines concur at D = tH. Hence the 4 perspectors, being duals of these lines, all lie on the dual of D, called the Polar axis.

Figure: The four Mineur lines m_x, each generating an hyperbola through H, converge on D = tH. The perspectors P_x of the conics are the duals of these lines and are hence collinear on ~D, the polar axis. The perspector P_o is off the page.

2. Equation of the original Feuerbach hyperbola is the isotomic conjugate of the Mineur line m_o.

a(b–c)s_a/x + b(c–a)s_b/y + c(a–b)s_c/z = 0

The other hyperbolas are extraversions of this equation.

The Feuerbach hyperbola is a weakened Jerabek hyperbola and shares many of its properties.

3. Special points and conic type

These Mineur conics, generated as the isotomic conjugate of the Mineur lines m_x = N_x—G_x, contain the points G_x, N_x, mG_x = M_x, and mN_x = I_x. Since D = tH is on all the Mineur lines, H is on all the conics, making all 4 conics rectangular hyperbolae.

The x-version of a point is on the x-version of the hyperbola.
The tripolars of points on the conic meet at the perspector.
More points on the conic are given below.

4. The Feuerbach hyperbolas and the N_x-affine parallelogram. The four points G_x, N_x, M_x, and I_x are both shared by the affine parallelograms of N_x and the Feuerbach hyperbolas. The Mineur lines G_x—N_x, form one diagonal, while the other goes through G and F_x, the hyperbola centers.

5. The asymptotes a₁ and a₂ are the Simpson lines of the meets of O—I_x with the circumcircle. The directions of the asymptotes are the isotomics of the meets of G_x—N_x with the Steiner ellipse, which are

or the Simson lines of the meets of O—I_x with the circumcircle.

The asymptotes are isotomic lines and their duals figure importantly in item 10.

The asymptotes of the Feuerbach hyperbolas are parallel to the axes of the I_x and M_x circumconics, and the Mandart inconic.

The equations of the asymptotes are

[ :b(c – a) s_b (c a (–a b³ + b⁴ + a³c + ab²c – b³c – a²c² + ac³) ± b(a² – bc+c²–ab) Z)² :]
where Z = √(abc (abc - 8 s_abc))

Axes: The axes are bisectors of the asymptotes and meet the axes of the above 3 conics at a 45° angle.

5. The centers are concyclic: F_x = mtd P_x = (: (c–a)² s_b :); all lie on the nine point circle. The centers go through the duals of the asymptotes.

6. The tangents at the vertices are the ex-Cevian lines of the perspector. The tangents at I_x are the lines O—I_x.

7. The four hyperbolas are tangent to each other and to the Lucas cubic at H. 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.

8. The dual of each hyperbola is an inconic with perspector tP_x and center mP_x. Since the Feuerbach's all go through H, these are all tangent to the line ~H, the dual of H. Since the P_x are collinear, so are the mP_x, the centers of the dual conics.

Figure: The dual Feuerbach conics, perspectors tP_x, all hyperbolas, are shown in blue. Black is the dual of H to which they are all tangent. The centers of the inconics mP_x lie on the medial Polar axis, shown red.

9. The switched circumconic, which has center and perspector switched, has perspector F_x = mtd P_x and center P_x. These hyperbolas have the line pairs ~G_x and ~N_x as asymptotes.

10. The last two conics meet the dual of the hyperbola center F_x at two isotomic points ~a_1x and ~a_2x, which are the duals of the hyperbola asymptotes. The lines from G to these points are tangent to the dual inconic. The dual of the center also goes through t(∞•~G_x) and t(∞•~N_x), the conjugates of the directions of the duals of the Gergonne and Nagel points.

The switched hyperbolas are also Mineur conics, generated by ~F_x which goes through the Feuerbach asympote duals, which are isotomic. The means that the medials of these two points are on these conics.

Figure: Original Feuerbach hyperbola, the switched conic and the dual inconic.
The original version of the Feuerbach hyperbola is shown in red. The blue circumconic is the switched conic. The other light blue one is the dual inconic. The blue conics meet at two points which are the duals of the Feuerbach asymptotes a₁ and a₂ and are on the ~F_o line, which is the dual of the hyperbola center. The affine parallelogram is shown in yellow. The duals of G_o and N_o are the asymptotes of the switched conic.

11. 4th intersections: The fourth intersection of the o-conic with the circumcircle, :b/(b²–ca): , is the isogonal of the Mineur endpoint. The Steiner intersection is the isotomic of the Mineur endpoint (X2481_o) = :1/b(b²–ca): . The fourth intersection of the hyperbolas with eachother is H.

12. The tangents at the intersections of G—F_x with the correcponding Feuerbach hyperbola is parallel to the Mineur line G_x—N_x.

The Feuerbach hyperbola organizes the weak conics

A magical, almost mystical, property of a Mineur conic, such as the Feuerbach hyperbola, is that it organizes a great many related conics whose perspectors and centers are on this hyperbola, and whose axes are parallel to the Feuerbach asymptotes.

These four points, all on the Feuerbach hyperbola, are related in the following way. Going from left to right we use the standard affine operations to generate the four points.

I_x — d — N_x — t — G_x — m — M_x.

These four points carry an amazing amount of structure. Each of the 6 pairings of these points expresses a conic relationship. There are three lines connecting them all, each of which generates a significant conic, or set of conics. The other three pairings are of conic center to conic perspector, which happens for 4 conics. This is summarized in the following diagram.

Figure and Chart: the 6 point pairs from the four points on the Feuerbach hyperbolas, showing the conic relations for each pair. The arrows represent operations indicated by the red names. The strong point on the weak lines defined by the two point is indicated lightly. The fact that K is on the I_x—M_x lines means that the axes of both are parallel. The fact that D is on the G_x—N_x lines means that all four hyperbolas will be rectangular.

Point pair	conic	comments
Gx, Nx on Mineur line Gx—Nx	Feuerbach hyperbola, the Mineur conic of the system	perspector is dual of Mineur line tripolars of points on hyperbola go through perspector.	The Mineur conic
Mx, Gx on line Mx—G—Gx	Yff hyperbola Yff parabola	perspector at infinity, center on Steiner inellipse perspector on Steiner ellipse, focus on CC, directrix H—Mx	Anallagmatic pair
Ix, Nx on line Ix—G—Nx	Nagel hyperbola Nagel parabola	perspector at infinity, center on Steiner inellipse perspector on Steiner ellipse, focus on CC, directrix H—Ix
Mx, Ix	Mx and Ix circumconics	Axes of both parallel to Feuerbach axes. 4th intersection dFo, where Fo is the Feuerbach center.	switched pair
Nx, Mx	Mandart inconics	Axes parallel to Feuerbach, goes through Fo, the Feuerbach center	Mated Pair
Gx, Ix	Incircle	Axes parallel to Feuerbach, goes through Fo, the Feuerbach center

Lines between the Feuerbach 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 Feuerbach hyperbola G_o and N_o = tG_o each give 2 such lines with two pairs of 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 Feuerbach conic, the isotomic points are D and H. The four Feuerbach 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. Not only do the antipodal lines go throught he 4 special points on the hyperbola, but so do the anallagmatic pair, the Yff hyperbola and the Nagel hyperbola.

Notation.Figure: The very interesting picture shows the relation between the Feuerbach hyperbola with its four important points to the anallagmatic pair of hyperbola (the Nagel and Yff) and special points on the Steiner ellipse and circumcircle that are related to the directions of the duals of G_o and N_o.
The parabolas associated with the Yff (center 1086_o) and Nagel (center 1146_o) hyperbolas are shown along with their foci, axes, and directrices.

Figure: The incircle, whose perspector G_o and center I_o are on the Feuerbach hyperbola and the Mandart inconic, whose perspector N_o and center M_o are also on the hyperbola, meet at F_o the hyperbola center.

The conics represented in this diagram represent most of the the well studied weak inconics:

Figure: Feuerbach and associated conics The Feuerbach hyperbola (red) with 2 circumconics (the M_o and I_o, each the center of the other) and 2 inconics (G_o and N_o, centers I_o and M_o). The inconics meet at F_o the Feuerbach center. The axes of these circumconics and inconics are parallel to the Feuerbach asymptotes.

Go here for a method of construction of these structures.

Points on the Feuerbach hyperbolas

The Feuerbach hyperbola is an example of a very special type of conic that I call a Mineur conic, which is a conic that is the conjugate of the line between a point and its conjugate. Here the conjugation is isotomic, but the above construction is general for Mineur conics using other conjugations.

Here are the 4 Hyperbolas with many of their points.

Figure: the Feuerbach hyperbolas with points on them.

[these points are from Peter Moses (thanks Peter!)]
Feuerbach circum-hyperbola, center 11, perspector 650 {1,4,7,8,9,21,79,80,84,90,104,177, 256, 294, 314,885,941,943,981,983,987,989, 1000,1039,1041,1061,1063,1156,1172, 1251, 1320,1389,1392,1476,1896,1937, 2298, 2320,2335, 2344,2346,2481,2648,2997,3062 ?,3065 ?}
Red point are weak. The blue point is strong, the single strong point on any of these conics. The green is octile. Cyan is super-weak. The points in pink are of lesser importance and are not discussed below. For other conics the points I ignore are seemingly irrelevant, what I call "silly" points. There are no such on Feuerbach. I have ignored the lightened ones just for my own convenience.

1251 = {b/(s_ca+√(3) ∆)} This is an octile point that comes in 8 versions. See here about X1251.

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

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.

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 ∞•~Io	9 b sb Mo the Mittenpunkt	71 b2(c+a)SB Mo the Mittenpunkt
519 c+a–2b ∞•(G—Io)	? (c2-a2)/(c+a–2b) ∞•(G—Io)	88 b(c-a)sb/(c+a–2b) 1, 100	? b2(c2–a2)SB/(c+a–2b) 1, 100
513 b(c–a) twS = ∞•~tIo	321 (c+a)/b twS = ∞•~tIo	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
? b(c2+a2–ab–bc) ∞•(Go—No)	? (c2-a2)/b(c2+a2–ab–bc) -	885 (c–a)sb/(c2+a2–ab–bc)	? b(c2–a2)SB/(c2+a2–ab–bc)
? b(c-a)sbb, ∞•~Go2	1446 (c+a)/bsbb,	7 1/sb, Go	1439 b(c+a)SB/sbb, -
? (c–a)sb(b2–ca)	? (c+a)/sb(b2–ca)	256 b /(b2–ca) 9, 43	? b2 SB(c+a)/sb(b2–ca) 4,256
521 b(c–a)sbSB	? (c+a)/bsbSB	4 1/SB H	65 b(c+a)/sb HGo
? (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
? (c–a) sb (c2+bc+ab+a2)	? (c+a)/sb(c2+bc+ab+a2)	2298 b/(c2+bc+ab+a2) -	? b2(c+a)SB/(c2+bc+ab+a2) 4, 2298
? (c–a)(c2+a2+ca)	? (c+a)/(c2+a2+ca)	2344 b sb/(c2+a2+ca) 1, 32	? b2(c+a)SB/(c2+a2+ca) 4, 2344
? 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
? (c2-a2)(c2+a2–2b2) ∞•~190	671 1/( c2+a2–2b2) Steiner intersection	? b sb/(c+a)(c2+a2–2b2)	? b2 SB/( c2+a2–2b2) Steiner intersection
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		b(c–a)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–c2a2+a4–b4) -	? 1/(c4–c2a2+a4–b4) gvG		? b2SB/(c4–c2a2+a4–b4) -
		Fissile points
		1251 b/(sca+√3 ∆) 1, 15 octile
	Fn
		177

The incenters of Cevian triangles of points on the Feuerbach hyperbola

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

It follows that the points which have an in/excenter of their cevian triangle at G_o must lie on the Feuerbach hyperbola. This can be generalized for any other point. 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_Go is a vertex of the G_o-Cevian triangle. I_aGo is one of the 4 incenters of this triangle, which is on the Jerabek hyperbola.

Figure: The blue hyperbola is Kiepert, red Jerabek, and the green one is Feuerbach. Notice that corresponding points from the above table are collinear with H.

II. What we know about quartile points

Quartile points such as those the Gergonnes, the Nagels, the Mittenpunkts, and many others come in 4 versions. Often they form desmic systems whereby 2 quartile sets are part of a maximally interconnected set of 16 lines and 12 points (4 lines per point, 3 points per line). These are beautiful constructs and one of the great pleasures of quartile points. The 4 Gergonnes and 4 Nagels form such a pair. These desmic systems reside on cubic curves. Here is the Gergonne-Nagel desmic system, displayed as a projected cube, residing on the 4 Feuerbach hyperbolas (blue) and the Lucas cubic (red). All 12 points are on the cubic while each weak point lies on the corresponding hyperbola.

For the Feuerbach hyperbolas this has the consequence that each of them shares ABCH and one each of the set of Nagel and Gergonne points with the Lucas cubic as shown in the picture.

Figure: The blue curves are the 4 Feuerbach hyperbolas. The red one is the Lucas cubic. The solid and dased black lines are the projected cube formed by the 12 points of the Gergonne-Nagel desmic system below, all on the cubic. This is a part of the group table for this cubic.

Gergonne-Nagel Desmic system
D (= tH = dK)	A	B	C
Go	Ga	Gb	Gc
No	Na	Nb	Nc
Harmon: Ro = tIo

O—I_x lines are tangent at I_x

These lines are shown in red in the following picture.

 

Relation to cubics
The G_x-N_x desmic system are a part of the Lucas cubic. It is known that a Mineur conic is tangent to a pivotal cubic at the conjugate of the pivot point. For the Lucas cubic, this point is H, so all four hyperbolas are tangent to eachother and the cubic at H. Each hyperbola intersects a cubic 6 times, all real in this case, so that there are 24 intersections. We have found all of them (4 N_o, 4 G_o, 4H, 12 ABC). The following picture shows the Lucas cubic, the four Feuerbach hyperbolas as well as the two strong hyperbolas, the Jerabek and Kiepert ones. Both of these are tangent to the Lucas cubic, the Jerabek at H and the Kiepert at G.

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 seen 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 original Feuerbach hyperbola ABCI_oG_o, this Cevian conic goes through the points A_Go B_Io C_Go A_Io B_Go C_Io. 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 Feuerbach ABCI_oN_oG_oM_o hyperbola, computation gives these points and lines, taking all combinations of the I_o,N_o,G_o,M_o for P and Q. These are shown in the following chart of charts. The I_o,M_o and N_o, M_o tables are the most interesting. Entries not filled in have complicated formulas.

This chart refers to the original Feuerbach hyperbola; of course an analogous one applies to each of the others.

	Ix	Nx	Gx	Mx
Ix	–	the co Pascal G–Io Steiner Kirkman Plucker	the co Pascal [:(c–a)sb2:] Io—Go Steiner Kirkman Plucker	the co Pascal [ : (c–a)/b: ] A-median Steiner gwT at infinity centroid Kirkman Mo (:(c–a)sb2–:) Plucker [:b(c–a):] through G ∞
Nx		–	the co Pascal [:b(c–a)sb:] Steiner Kirkman Cayley Plucker	the co Pascal [:(c–a)/sb:] Steiner (:(b2–ca)sb:) (:sb/(c+a):) Kirkman (:b(c+a)sb:) (:b(c–a)(b2–)sb:) Plucker [:(c+a)/sb:]
Gx			–	the co Pascal G—Go Steiner Kirkman Plucker
Mx				––
	Ix	Nx	Gx	Mx

Locus

From Bernard Gibert in FG??: Proposition 6 ([5, 2, p.551]). The Mandart triangle T(t) and ABC are orthologic. The perpendiculars from A, B, C to the corresponding sidelines of P_a P_b P_c are concurrent at Q_t = a /(aS_A + 4∆t) : · · · : · · · .
As t varies, the locus of Q_t is the Feuerbach hyperbola.