The Kiepert Hyperbola, Parabola, and Circumparabola
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 the conic degenerates into its asymptotes, and its real foci become the imaginary ones and vice-versa. This explains why an hyperbola has 4 foci. Also of interest is the motion of the points on the hyperbola itself. This movie (2M) is formated for the iPod. A larger one (30 M.) is here. It may take awhile to load but you can see much more detail.

Contents Kiepert and dual conics Properties Points Points on rectangular hyperbolas Conic variables Tangents 9-Point Circle, the invariant conic Degeneracies (with movie) Locus definition Pascal's Theorem Generalized Kiepert conics Link to GSP file, begun by Peter Moses with much added by me. This file contains tools for the relevant ETC points, with all extraversions. Movie in iPod .m4v format. Peter Moses has contributed significantly to the section of points on the Kiepert hyperbola.

The Kiepert hyperbola and Kiepert Parabola were only known to a few specialists (they are mentioned in neither Johnson or Altshiller-Court, two standard references) until the paper The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle by Eddy and Fritsch. Interestingly this paper was next to the paper where Clark Kimberling's Central Points and Central Lines in the Triangle. These papers, particularly the latter, got me interested in triangle geometry.

Eddy and Fritsch began this way:

If a visitor from Mars desired to learn the geometry of the triangle but could stay...only long enough for a single lesson, earthling mathematicians would, not doubt, be hard pressed to meet this request....we have an optimum solution to this problem. The Kiepert conics, though seemingly unknown today, constitute a significant part of the geometry of the triangle and to study them one has to deal with many fundamental concepts related to this geometry such as the Euler line, Brocard axis, circumcircle, Brocard angle, and the Lemoine line in addition to well know points including the centroid, circumcenter, orthocenter, and the isogonic centers.

For me these articles and this hyperbola marked my beginning in triangle geometry. I have come back to the Kiepert hyperbola many times, at first to doubt the claims made for it; more recently to celebrate this wonderful object. I have found a new way to think about this hyperbola and parabola, a way that places it firmly in relation to other points, lines, and conics.

A listing of notation is found here.

Kiepert hyperbola, Kiepert parabola, and a circumparabola

Figure: The Kiepert hyperbola. This picture shows the hyperbola in blue with its center, foci (green, red text), asymptotes, together with significant points, lines and circles. [Thanks to Peter Moses for his implementation and computation of the Kiepert foci.] The two Steiner elllipses are shown in black. The circumcircle, 9 point circle, and the auxiliary circle are light black. Strong points are shown in blue; 3-fold points such as the triangle vertices in yellow; 2-fold points, such as the foci, in green; and 4-fold points, such as the Spieker points in red. S_1,2 are the intersections of G—K with the Steiner ellipse. Q_1,2 with the circumcircle. See point notations below. The blue parallelogram connects G, H, and the hyperbola intersections with the circumcircle and the Steiner ellipse. Its directions are the Euler lines of ABC and the image Brocard triangle (the first Brocard triangle).

Discovered by Kiepert using a locus argument, we will, initially, present the Kiepert hyperbola in a different way, as an immediate deduction of the affine theory of the triangle.

Notation is here.

Consider the symmedian point K. There are two affine invariant lines associated with it: G—K and ~K, the dual of K. These lines are invariant in that, after an affine tranformation, K becomes P, say, G—K becomes G—P and ~K becomes ~P. Most of the properties listed below are invariant in the same sense.

The dual of a point is a line, the tripolar of its isotomic conjugate. The isotomic conjugate of a line is a circumconic with perspector the dual of the line. The line ~K produces the circumcircle, whose perspector is K, and the line G—K produces the Kiepert hyperbola, whose perspector is tS, the isotomic conjugate of the Steiner point, which is at infinity. The schema below, part of the much larger schema of strong conics, shows the relationships.

This diagram shows a special set of points related to K, and thereby a special, interrelated set of conics: D (= dK), H (=tdK), O (=mtdK). Each of these points implies a special conic relationship to the others. K and O are related as perspector and center of two conics. K and H = tdK are the center and perspector of an inconic. O and D = tdO are the center and perspector ao another inconic. OH and KD are lines through G and define conics with perspectors at infinity. HD is a Mineur line and defines the Jerabek hyperbola on which these four point lie.

Figure: The m-t-d relationships.

Almost all of the properties listed below are shared by any other circumconic with perspector at infinity (ie, derived from a line through G). The exceptions are noted in the item listing and are two: the fact that ~K is perpendicular to the Euler line and the ~S is perpendicular to G—T. the Euler line of the Brocard image triangle (aka the First Brocard triangle).

1. Definition: The Kiepert Hyperbola is the isotomic conjugate of the line G—K. Its perspector is tS = (: c²–a² :), the dual of G—K and the infinite point on ~K. The notation tS denotes the isotomic conjugate of the Steiner point. The equation of the Kiepert hyperbola is

(b²–c²)/x + (c²–a²)/y + (a²–b²)/z = 0.

The Kiepert hyperbola is also the isogonal conjugate of the Brocard meridian line O—K. Kiepert gave a locus definition (below) which is still the traditional definition. The affine approach used here derives a great number of properties rapidly, with little effort, as well as putting this hyperbola in a larger context of triangle conics.

2. This is a rectangular hyperbola: The line G—K intersects the Steiner ellipse twice, isotomic conjugate of which is at infinity, so this conic is an hyperbola, where those infinite points are the directions of the asymptotes. The line G—K goes through D = tH = dK so that the Kiepert hyperbola goes through H. Either of these facts guarantee that this is a rectangular hyperbola. The Kiepert hyperbola is the only rectangular hyperbola with perspector at infinity.

The asymptotes are also the Simpson lines of the intersections of O—K with the circumcircle, true for rectangular hyperbolas only. The endpoints of the asymptotes are the conjugates of the intersections of G—K with the Steiner ellipse.

The intersections of G—K with the Steiner ellipse are ( : c²+a²–2b² ± Q : ) where Q = √(a⁴ + b⁴ + c⁴ – b²c² – c²a² – a²b²). Their conjugates are the endpoints of the asymptotes. The equations of the asymptotes are ( : (c²–a²)(c²+a²–2b² ± Q)² : ) which can be shown to be conjugate.

The easiest construction of the asymptotes is this: Draw the lines from mS, the center, to the intersections of G—K with the Steiner inellipse.

3. Axes: The axes are the bisectors of the asymptotes. They are parallel to the asymptotes of the rectangular hyperbola with perspector ( : c^4– + a^4– + 2b^4–: ) = mm K^2–, which is the intersection of the tripolar of S with the Polar axis.

4. The dual of a point on G—K is parallel to ~K, which is perpendicular to the Euler line. Hence the tripolar (the dual of the isotomic conjugate) of any point on the Kiepert hyperbola is perpendicular to the Euler line. (Note: it is generally true that the dual of a point on G—P is parallel to ~P, but only for the Kiepert hyperbola is this line perpendicular to the Euler line).

5. Center: The center for a circumconic is mtd of the perspector and, for the Kiepert hyperbola, it is M (= mS) = (:  (c²–a²)² :) which is on the Steiner inellipse. The center is also on the nine point circle since the centers of all rectangular hyperbolae are there.

~mS, the dual of the Kiepert center, goes through S and F and is tangent to the Steiner ellipse at S.

6. The Kiepert parabola: The dual of the Kiepert hyperbola is an inparabola (as is the dual of any hyperbola with perspector at infinity), the Kiepert parabola, whose perspector is the Steiner point S, the conjugate of the hyperbola perspector tS. Its focus is F = pS on the circumcircle. The axis is through F parallel to ~K, i.e., perpendicular to the Euler line and parallel to the polar axis. The directrix is perpendicular to ~K through H, which makes it the Euler line.

Since a conic and its dual never intersect, the Kiepert hyperbola and parabola do not intersect.

There is a nice formula relating a point on a circumconic to the corresponding point (the point of tangency) on its dual. Let the circumconic, perspector P = (l:m:n) have equation l/x + m/y + n/z = 0. Its corresponding inconic has equation √( lx ) + √( my ) + √( nz ) = 0

If Q = ( : y₁ : ) is on this conic then l/x₁ + m/y₁ + n/z₁ = 0. The dual of Q is the line x₁ x + y₁ y + z₁ z = 0. The point (: m/y₁² :) is manifestly on this line as well as the inconic. So this must be the point of tangency. Hence the transformation : y : –> : m/y² : takes a point on a circumconic to the corresponding point of tangency on its dual inconic. For the Kiepert hyperbola to the Kiepert parabola this is y -> : (c² – a²)/y² : As an aside (: m/y₁ :) is on the line at infinity.

7. The switched circumparabola: The center and perspector of a circumconic can be switched, the center becoming the perspector and vice-versa. This circumconic is a circumparabola with perspector M = mS on the Steiner inellipse, and center tS, at infinity. Like the Kiepert Parabola, its axis is also in the direction of ~K, the point tS, and is perpendicular to the Euler line.

Figure: The Kiepert circumhyperbola; its dual, the Kiepert inparabola; and the switched conic, a circumparabola. The dual of the hyperbola center goes through the intersections of the parabolae. The duals of the intersection points are the hyperbola asymptotes. The lines from G to these points are tangent to the Kiepert parabola. The axis of the Kipert parabola is parallel to ~K and perpendicular to the Euler line, which is its directrix.

8. Duals of the Kiepert asymptotes. The inparabola (the dual) and the circumparabola (the switched) meet in two isotomic points on ~mS. The duals of these two intersections are the Kiepert asymptotes a₁ and a₂.

The switched circumparabola is the conjugate of the line ~mS, the dual of the Kiepert hyperbola center. Since ~mS goes through ~a₁ and ~a₂, the duals of the asymptotes (isotomic, remember), the switched circumparabola is a Mineur conic, so that the four points ~a₁, ~a₂, m~a₁ and m~a₂, are all on it.

The lines G — ~a₁ and G — ~a₂, are tangent to the inparabola.

9. Fourth intersections: The fourth intersection of the Kiepert hyperbola with the circumcircle is T, the Tarry point, the isotomic of the intersection of ~K and G—K (note that this gives a new definition for the Tarry point). Its fourth intersection with the Steiner ellipse is U = nS = ( : 1/(c²+a²–2b² : ), the reflection of S in G and the isotomic of the infinite point on G—K.

The fourth intersection with all rectangular hyperbolas, including Jerabek and Feuerbach, is H. The fourth intersection of all hyperbolas with perspectors at infinity, including the Eulerian hyperbola and the Yff hyperbola, is G.

The axis of the inparabola meets the circumcircle at its 4th intersection with the circumparabola.

10. ~S is tangent to Steiner inellipse at the Kiepert center M = mS and goes through J, the Jerabek center. It is perpendicular to G—T, which shows that the projection of mS onto G—T is J.

~mS, parallel to ~S, is tangent to the Steiner circumellipse at the Steiner point S and goes through F = dJ.

11. GHUT. John Conway pointed out that the points GHUT form a parallelogram where T, the Tarry point, is the intersection of the Kiepert hyperbola and the circumcircle, while U = nS is the intersection of the Kiepert hyperbola and Steiner ellipse. The sides of this parallelogram are parallel to the Euler line (perpendicular to ~K) or to G—T the image Euler line, the Euler line of the Brocard image triangle (Brocard's first triangle), which is perpendicular to ~S. (Note: this parallelogram is general, but that the directions of the sides are related to the two Euler lines is unique to the Kiepert hyperbola).

ASCII picture by John Conway announcing the GHUT parallelogram. [now the words of JHC]: "The four vertices T,U,G,H of this parallelogram all lie on the Kiepert Hyperbola, whose center is mS. As you'll see, this implies that U is the reflection of S in G, and since S lies on the Steiner ellipse, whose center is G, we see that U must also lie on the Steiner ellipse. Now S is traditionally defined to be the 4th intersection (after A,B,C) of the Steiner ellipse and the circumcircle, while the Tarry point T is the other end of the diameter through S. So T is the 4th intersection of Kiepert Hyperbola with the circumcircle, and U its fourth intersection with the Steiner ellipse. "

Figure: the Kiepert hyperbola is in red, the Steiner ellipse and the circumcircle in blue. The GHUT parallelogram is in black. The sides of this parallelogram are parallel to the Euler line and the image Euler line. The Evans conic is in black and goes through F_x, I_x, and N_x (where x is one of the extraversion indices n, s). The I_x—F_x lines connecting the isodynamic points to the Fermat points are tangent to the hyperbola at F_x. The line through mS, the Kiepert center parallel to G—T goes through the two Fermat points.

12. Tangents at important points: G—K is tangent to the Kiepert hyperbola at G. The line parallel to G—K through nS is tangent at nS. This line meets the line parallel to ~K through S on the Steiner ellipse at the barycentric product of S and nS. The circumparabola goes through this point.

The line K—mS (parallel to G—T) intersects the Kiepert hyperbola at the Fermat points. The tangents F_n—I_n and F_s—I_s at these points are parallel to the Euler line.

Tangents at A, B, C are the ex-Cevian lines of the perspector, but see below for another nice construction of the tangent lines.

A second way to get the tangents is from the point tables given directly below. A third way is by linearizing at the point of tangency. A fourth way is Pascal's Theorem.

point	tangent line	line coordinates
G	G—K	[ : (c2–a2) : ]
H		[ : SBB (c2–a2) : ]
Fn,s	Fn—In line and Fs—Is line, parallel to Euler.	[ : (SB±Sp/3)2(c2–a2) : ] parallel to Euler
A	C-edge of preCevian triangle of Kiepert perspector A—(e•CH—BG•CG—BH)	[ 0 : (a2–b2) : (c2–a2) ]
B	B-edge of preCevian triangle of Kiepert perspector B—(e•CH—AG•CG—AH)	[ (a2–b2) : 0 : (b2–c2) ]
C	C-edge of preCevian triangle of Kiepert perspector C—(e•BH—AG•BG—AH)	[ (c2–a2) : (b2–c2) : 0 ]

13. The only point in the plane that subtends a right angle to the Kiepert Hyperbola is the center mS.

Points on Kiepert hyperbola and Kiepert Parabolas

The chart below shows points and tangent lines on the Kiepert circumhyperbola and its dual, the Kiepert parabola. Strong points and lines are on a blue background and weak ones on a red one. I find the points on these by taking the most relevant points at infinity, then moving them to the Kipert hyperbola.

Here is Peter Moses' compilation of ETC points on the Kiepert Hyperbola. (I have colored them: blue for strong, red for weak, green for fissile; I have greyed out one that seem to me to be worth ignoring.) The purple are centers of similarity of the Brocard circle with various circles, they have a factor of √( S₂₂ = sum a²b²).
[Peter Moses] Kiepert circum-hyperbola, center 115, perspector 523
{2,4,10,13,14,17,18, 76 (R=tK), 83 (tmK), 94 (1/b²(S_B²-c²a²), 96, 98 (1/(c⁴+a⁴–a²b²–b²c²), 226, 262,275, 321, 485,486 (Vecten) , 598 (1/(2c²–b²+2a²), 671 (1/(c²–2b²+a²), 801,1029,1131,1132,1139,1140 (pentagon pts), 1327 (b(c-a)/(c²+a²–bc–ab), 1328, 1446,1676,1677,1751,1916 tK^2– 1/(b⁴–c²a²),2009, 2010, 2051,2052 (sec 2B), 2394, 2592,2593,2671,2672,2986,2996}

Dual Keipert = Kiepert parabola, "center" 523, perspector 99
{523 tS, 669 pgS {a⁴(b²-c²)},1649 {(b²-c²)(-2 a²+b²+c²)²}, 2528 {(b²-c²) (b²+c²)²}}

115 circumconic (swithched), center 523, perspector 115
{476 {1 / (b² - c²) (4S_A²-b²c²)}, 523, 685,{a²/(a⁴⁺)} 850 fF, 892 (4th SE), 2395 {(b²-c²)/(b^4–+ c^4–) }, 2501 {(b²-c²)/S_A}}

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

In the following table

Dualizing

Point on Kiepert hyperbola	Point on Kiepert parabola
Tangent to Kiepert parabola	Tangent to Kiepert hyperbola
: y :	:(c2–a2)/y2:
G	523 tS = ( : c2–a2 : ), its infinite point
H	: SBB (c2–a2) :
R = tK 4th tan to inner SE	669 : b4 (c2–a2) :
:(c2–a2)/(c2+a2–2b2):	(:(c2+a2–2b2)2/(c2–a2):)
So	(: (c–a)/(c+a) :)
T = tmK2– : 1/ (c4–+a4–) :	: (c2–a2)(c4–+a4–)2 :
U = nS :1/(c2–2b2+a2)::)	1649 : (c2–a2)(c2–2b2+a2)2 :
321o (:(c+a)/b:)	: b2(c–a)/(c+a) :
: 1/b2SB2 :	: b4SBBBB (c2–a2) :
tmK :1/(c2+a2):	2528  : (c2–a2)(c2+a2)2:
1916 tP– :1/(b4 – c2a2):
226o (c+a)/(c+a–b)	: (c–a)(c+a–b)2 :
(c+a)/(c+a–2b)	: (c–a)(c+a–2b)2 :
1/(SB ± Sq) inclues Fermat (q = p/3), Napoleon (q = p/6), and Vecten (q = p/4) points. See locus discussion below.	: (SB±Sq)2 (c2–a2) :
1/(a2 + constant)	: (a2 + const)2 (c2–a2) :

Points on rectangular hyperbolas

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.

H, the orthocenter is the fourth intersection of any two rectangular hyperbolas. The fourth intersection is a projection center for the projective transformation that takes one hyperbola into the other. The line through corresponding points goes through the fourth intersection. Algebraically this is implemented by the barycentric product that takes one perspector into the other.

The following picture shows six rectangular hyperbolas: the Kiepert, Jerabek, and all four Feuerbach hyperbolas, and the ETC points that lie on them. The points arose as a collaborative effort between Peter Moses and myself. All the points lie on lines through H.

The algebraic projection from Kiepert to the Feuerbachs is barycentric multiplication by H_x, one of the four Shiffler points. The projection from Kiepert to Jerabek is multiplication by O, the circumcenter.

Points listed as 71_x (where x is one of o,a,b,c) represent a version of the quartile point X71, and similarly. 177_no represents the two indices of of an octile point. The 8 versions of this point, which are projections of F_n and F_n, the Fermat points, lie on two lines 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 So	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) -	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 -	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, -
521 b(c–a)sbSB	? (c+a)/bsbSB	4 1/SB H	65 b(c+a)/sb HGo
		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
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)(c^4-c^2 a^2+a^4-b^4) -	? 1/(c^4-c^2 a^2+a^4-b^4) gvG		? b2SB/(c^4-c^2 a^2+a^4-b^4) -
		Fissile points
		1251 b/(sca+√3 ∆) 1, 15 octile
	Fn
		177

Figure: 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 from the above table on the six hyperbolas.

Kiepert conic variables and degenerate behavior

Peter Moses has computed the radius of the auxiliary circle which is

[PM] Q² = a² + b² + c² – a² b² – b² c² – c² a² , an old favorite, related to distance circumcenter to symmedian point.
T = (a² – b²) (b² – c²) (c² – a²)
S = twice area ABC . S_A = (b² + c² – a²)/2 & et. seq.
The squared radius of the "incircle" (auxiliary circles) of the Kiepert circumhyperbola is
S T / (4 Q³)
 

Here are the conic variables for this hyperbola. The conic variables are a, semi-major axis; b, semi-minor axis; c, semi-focal length; d, radius of director circle.

The Kiepert hyperbola
perspector	tS = ( : c2 – a2 : )
center	mS = ( : (c2 – a2)2 : )
semi-major axis	√(S T / (4 Q3))
semi-minor axis	√(S T / (4 Q3))
semi-focal length	√(S T / 2Q3)
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 when the hyperbola degenerates, becoming a median and an edge.

Tangents at A, B, C.

The Kiepert tangents at A, B, C form the pre-Cevian triangle of tS, the Kiepert perspector. A nice way to construct them is to do the following. The lines A_H—C_G and C_H—A_G meet where the Euler line intersects the B tangent, which gives a construction for that tangent. Note: B_G is Conway notation for the B-trace of the centroid. B^tS is a vertex of the preCevian triangle of tS.

A_H B_G C_H A_G B_H C_G forms a Third hexagon which has 3 double intersections with the Euler line at the intersections of the Euler line with the Kiepert asymptotes.

Figure: The tangents at A, B, C are shown in relation to an inscribed hexagon the the Third type. The Euler line is bold and red. The tangents are dashed and red. The Kiepert hyperbola is bold and blue. The asymptotes meet at the vertices of the tS pre-Cevian triangle, whose perspectrices are perpendicular to the Euler line. A^tS is Conway notation for the pre-Cevian vertex of tS.

The 9-point circle, the invariant conic for the Kiepert hyperbola
Central points and lines from the Hexagrammum Mysticum of the 9-point circle.

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 Kiepert hyperbola ABCGH, this conic goes through the points A_H B_G C_H A_G B_H C_G, which is the 9-point circle. John Conway has pointed out that these points, taken in this order, form a very special hexagon which is invariant under permutations of ABC (also interchanges of G, H), 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 points and lines by the names "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. These points lines and the invariant hexagon are shown in the chart and picture below.

A good name for this conic is the invariant conic of G and H.

For the Kiepert ABCGH hyperbola, computation gives these points and lines.

In the following table, q is the base angle of the similar isosceles triangles. Here A' is the complementary angle to A, and W is the Brocard angle.

Pascal's Theorem for the Kiepert hyperbola
how the Kiepert Hyperbola organizes many points in the triangle plane.

This section is based on a monograph by Jean Peyoux, Contribution a la Geometrie Moderne du Triangle:

There are 5 particularly important points on the Kiepert hyperbola: A, B, C, G, H. If we use these 5 points in Pascal's theorem, the edges of the Pascal hexagon will be lines defined by pairs of these points. Their intersections will, three at a time, create the Pascal lines. In this way miscellaneous triangle points, such as the intersection of a median with an altitude, are organized in relation to well known triangle structures. Since there are 60 Pascal lines, many points will be organized.

Pascal's theorem needs six points so we double up one of the above 5 so that Pascal's theorem will use the tangent line from the doubled point. In this case there are 60/5 = 12 lines. We will start by counting A twice. (more details later). The following picture shows three Pascal lines converging to a Steiner point.

This Steiner point is the concurrance of Pascal lines defined by hexagons AAGHBC, ACGABH, and AHGCBA, as explained here. The hexagon AAGHBC generates a Pascal line as the line connecting the intersection of opposite edges AA.HB (meet of A-tangent and B-altitude), AG.BC (meet of a-edge and A-median), GH.CA (meet of Euler line with b-edge of triangle).

The Pascal structure of a circumconic organizes the structure of these intersections. The are not ordinarily centers.

Figure: Three Pascal lines converge to a Steiner point.

The picture below show how the Kiepert Hypebola organizes miscelleneous points via Pascal's theorem. The table, whose infomation comes from Peyroux's monograph Contribution a la Géométrie Moderne du Triangle, shows points on the tangent lines, any one of which could be used to construct the line.

Tangent line	points on tangent line, defined as concurrence of lines.
through C	e, BH—AG, BG—AH c, hc•ma—hb•mc, hc•mb—hb•mc, ma, BH—CG, hb, CH—BG, CG—e•c mb, BH—hc•ma, CH—e•c
through B	etc
through A	etc
through H	a, ha•mc—e•c, ha•mb—e•b b, hb•mc—e•c, hb•ma—e•a, CH—ha•mc, mb, AH—hc•ma ma, CH—hb•mc, BH—hc•mb c, hc•mb—e•b, hc•ma—e•a, mc, AH—hb•ma, BH—ha•mb
through G	a, hc•ma—e•c, hb•ma—e•b b, hb•ma—e•c, ha•mb—e•a, AG—ha•mb, hc, BG—hb•ma hb, CG—hc•ma, AG—ha•mc ha, hc•mb—CG, hb•mc—CG, c, hb•mc—e•b, ha•mc—e•a

Here h_a = altitude through A.
m_a = median through A.
e = Euler line.
a = A-edge.
A_G = A-trace of G = midpoint on A-edge.
A_H = A-trace of H = foot of A-altitude.

Figure: The intersections of Pascal lines used to determine the intersections on the C-tangent to the Kiepert hyperbola. ∆_xy is the intersection of the x-altitude with the y-median. a, b, g are the intersections of the Euler line with the triangle sides. This is the Hexagrammum Mysticum for this case.

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