The weakened Kiepert (Yff) hyperbolas and theYff Parabolas
and associated structures:

Contents Properties Points Conic variables (soon) Weakened Spieker pts Tangents Degeneracies (with movie) (soon)

These pages are dedicated to the properties of two semi-famous circumconics.

For each there is
1) a derivation of the conic based on affine principles. This is often not the traditional derivation.
2) a picture showing the conic and its relation to its dual inconic and switched circumconic, the circumconic with perspector and center switched.
3). a table showing points on the conics and lines tangent to it.
4). a table of triangle variables (soon).
5). a discussion of conic degeneracies with a movie (soon).
6) (eventually) a discussion of locus properties.

Notation is here.

Weakened Kiepert hyperbolae, Yff parabolae, and 4 other parabolas

Figure: The weakened Kiepert hyperbola and Yff parabola, original versions. This picture shows the hyperbola in blue with its center and asymptotes, together with significant points, lines and conics. The two Steiner ellipses are shown in black. Strong points are shown in blue; 2-fold points in green; and 4-fold points, such as the Spieker points, which are on G—I_o, not the hyperbola, in red. The I_o-circumconic and switched circumparabola are shown.

We will, initially, present the weakened Kiepert hyperbola as an immediate deduction of the affine theory of the triangle. Notation is here. This hyperbola is related to the Yff parabola but the hyperbola has never been named. I will choose to call it the Yff hyperbola or the weakened Kiepery hyperbola. They are both an essential part of the weak theory of circumconics.

Note: a strong object is weakened by the transformation { a² –> a, b² –> b, c² –> c}, and is only meaningful in barycentric coordinates. For example, incenter is the weakened symmedian point. A weakened point will often have extra versions. The great advantage of this operation is that affine structure is preserved For example here is a part of the affine structure of points, lines, and conics. And here is the weakened one. Different objects; same structure. A very powerful operation.

Consider the incenter I_o, where "o" is the original incenter, one member of a quartile set. There are two affine invariant lines associated with it: G—I_o and ~I_o, the dual of I_o. The isotomic conjugate of a line is a circumconic with perspector the dual of the line, hence these two lines naturally and affinely produce conics. ~I_o produces the I_o-circumconic, whose perspector is I_o, and G—I_o the weakened Kiepert hyperbola, whose perspector is X514_o = ( : c–a : ), the endpoint of ~I_o which is at infinity.

Note: "affine invariant" in this context refers to the stated relationships. The incenter as a point is not affine invariant, many of its relations to other lines and point are. In particular the relationship between a point P, its dual ~P, and the line G—P is invariant.

The dual of a point is a line, the tripolar of its isotomic conjugate.

These are weak (quartile) conics:
The properties listed here are shared by any other circumconic with perspector at infinity and its dual parabola. However since these are weak conics and come in four versions, there are properties due to the interaction of the four versions that are not general shared. The variable x = (o, a, b, or c) indexes the four versions.

Figure: The four weakened Kiepert hyperbolas. The duals of I_x are shown and indicate the directions to the perspectors, which are the endpoints of these lines. The centers C_x are on the Steiner inellipse.

1. Definition: The weakened Kiepert Hyperbolae are the isotomic conjugates of the lines G—I_x, or equivalently the isogonal conjugates of the lines K—pI_x. Its perspectors are 514_x, the duals of G—I_x and ∞•~I_o, the infinite point on ~I_x . ∞•~I_o = 514_o = ( : c–a : ), the isotomic conjugate of the weakened Steiner point. Here 514_o indicates the original version (of 4) for X514. The equation of the weakened Kiepert hyperbola (original version) is

(b–c)/x + (c–a)/y + (a–b)/z = 0.

The other formulae are extraversions of this one.
X514 is the notation for point found in ETC.

The affine approach used here derives a great number of properties with little effort, as well as puts this hyperbola in a larger context of triangle conics.

2. Asymptotes: The lines G—I_x intersect the Steiner ellipse twice, the isotomic conjugate of which is at infinity, so these conics are all hyperbolae, where those infinite points are the directions of the asymptotes. The asymptotes are the Simpson lines of the intersections of K—pI_o with the circumcircle.

The equations of the asymptotes are ( : (c–a)(c+a–2b ± q_x)² : )
where the q_x are the 4 versions of √(a²+b²+c²–bc–ca–ab)

3. The line G—I_x goes contains I_x and N_x = tG_x = dI_x so that the weakened Kiepert hyperbolae goes through tI_x and the Gergonne points G_x. A chart with more points is given below.

4. The dual of a point on G—I_x is parallel to ~I_x. Hence the tripolar (the dual of the isotomic conjugate) of any point on the weakened Kiepert hyperbola is parallel to ~I_x.
Note: the dual of point on G—P is parallel to ~P.

5. Center: The center for a circumconic is mtd of the perspector and, for the weakened Kiepert hyperbolae, the centers are 1086_x , the extraversions of 1086_o = (: (c–a)² :) the weakened Kiepert centers, all on the Steiner inellipse.

The duals of the centers are the four tangents to both the Steiner ellipse and the circumcircle.

6. The Yff parabola: The duals of the weakened Kiepert hyperbola are inparabolae with perspectors 190_o = t(∞•~I_o) = (: 1/(c–a) :), the weakened Steiner points, isotomic conjugates of the hyperbola perspectors. These are the 4 Yff parabolas, the weakened versions of the Kiepert Parabola.

The axis_o is [ : (c–a)² s_b (c^2–+a^2–) : ], which is parallel to ~I_o and goes through its focus.

Figure: The 4 Yff inparabolae, their axes, their perspectors (red), their foci, the weakened Kiepert Foci (green), and their directrices. The axes are parallel to one of ~I_x. The perspectors are the weakened Steiner points t(∞•~I_x), all on the Steiner ellipse. The line from a pespector to a focus goes through S, the Steiner point. The directrix of each parabola goes through H, which unfortunately, is near B for this triangle

7. Foci and directrix: The focus of the Yff parabolas is the "pro" operation of the perspectors. The pro operation takes the Steiner ellipse to the circumcircle; in this case it takes a perspector to a focus, 101_x the 4 versions of ( : b²/(c–a) : ). The line from the perspector to the focus of each parabola goes through S.

The directrix of the Yff parabola is perpendicular to ~I_o, and goes through the orthocenter. Its equation is

(b–c)S_A x + (c–a)S_B y + (a–b)S_C z = 0

7. The switched circumparabolae: The center and perspector of a circumconic can be switched, the center becoming the perspector and vice-versa. Since its perspector is on the inner Steiner ellipse, these circumconics are parabolas with perspectors 1086_x = mwS, and center at infinity 514_x. Its axis is in the direction of ~I_x, the endpoint of which is the perspector.

The two parabolas meet on ~1086_x, the dual of the hyperbola center, a mutual tangent. The duals of these two intersections are the corresponding hyperbola asymptotes. The lines from G to the two intersections are tangent to the particular version of the Yff parabola.

8. Fourth intersections: The fourth intersections of the weakened Kiepert hyperbola and extraversions with the circumcircle is wT, the weakened Tarry point and extraversions. These points are the isotomic conjugates of the intersections of ~I_x and G—I_x. Their fourth intersections with the Steiner ellipse are 509_x the extraversions of ( : 1/(c+a–2b) : ), the reflection of the weakened Steiner point wS in G, also the conjugate of the infinite point on G—I_x.

9. ~wS is tangent to Steiner inellipse at the weakened Kiepert center wM = mwS and goes through the Feuerbach point F_o = wJ, the weakened Jerabek center. More about this below.

10. The points G G_o wU wT form a parallelogram where wT, the weakened Tarry point, is the intersection of the weakened Kiepert hyperbola and the I_o-circumconic, while wU = nwS is the intersection of the weakened Kiepert hyperbola and Steiner ellipse. The sides of this parallelogram are parallel to the G—G_o line or to G—wT, the second axis of the affine parallelogram of G_o.

11. G—I_x is tangent to the weakened Kiepert hyperbola at G. The line parallel to G—I_x through nwS_x is tangent at nwS_x. This line meets the line parallel to ~I_x through wS_x on the Steiner ellipse at the barycentric product of wS_x and nwS_x. The circumparabola goes through this point.

12. The line I_o—mwS (parallel to G—wT) intersects the weakened Kiepert hyperbola at the weakened Fermat points. The tangents at these points are parallel to the weakened Euler line G—G_o.

13. Tangents at important points: Tangents at A, B, C are the ex-Cevian lines of the perspectors but see below for another nice construction of the tangent lines. The tangent at G is G—I_x

A second way is by linearizing at the point of tangency. A fourth way is Pascal's Theorem (below).

14. The director circle of the weakened Kiepert hyperbola is ??.

Figure: The weakened Kiepert circumhyperbola; its dual, the weakened Kiepert inparabola; and it 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 weakened Kiepert parabola. The axis of the weakened Kiepert parabola is parallel to ~I_o and perpendicular to the Euler line, which is its directrix.

The Yff conics in context

By the affine theory conics are governed by the Mineur conic of the appropriate type. In this case these are weak conics and are governed by the Feuergach hyperbola, as seen here. The diagram that organized the weak conics is

Points on the weakened hyperbola (the Yff hyperbola) and the Yff parabola

Peter Moses' compilation of ETC points on the weakened Kiepert hyperbola. The points are colored by type; red is weak, blue is strong, and green is fissile. Light red (or pink) are points that do not seem very important to me.

A,B,C,G,G_o. center 1086_o, perspector 514_o
{2, 7 (G_o), 27 1/(c+a)S_B, 75 (tI_o), 86 (tS_o), 234, 272, 273 1/bs_bS_B, 310 (tpS_o), 335 (1/b^2-) , 554, 1081 1/(r3 s_oa ± ∆), 673 (1/mb^2-), 675 1/(a³–a² b+c²(-b+c)), 871, 903 1/(c+a–2b), 1088 1/b s_b², 1223,1240,1246,1268 (1/(c+a+2b), 1440, 1659 1/(–a²+b²+2ca+c²+∆)}, 2296, 2400,2989}

Peter only found one ETC point on the Yff parabola

Peter Moses has studied the ETC points on the Yff parabola. He has found 2, one being the infinite point.
Yff Parabola "center" 514, perspector 190
{514 t(∞•~Io), 649 (pwtS)} pwS 101 is focus

The following table shows the points on the line at infinity, which have an origin as centers in the triangle plane, and their projection to the weakened KH and the Yff parabola. Each set of coordinates is considered to simultaneously represent a point (as point coordinates) on one conic and a tangent line (as line coordinates) to its dual.

notation

point at infinity	point on weakened KH	point on Yff parabola
tangent to SE	tangent to Yff parabola	tangent to Yff hyperbola
y	(c-a)/y	y2/(c-a)
	From the Incenter
514 c–a twS =∞•~Io	2 G	514 (c–a) infinite point direction of axis
519 c+a–2b ∞•(G—Io)	? (c–a)/(c+a–2b) -	? (c+a–2b)2/(c–a)
513 b(c–a) ∞•~tIo	75 1/b isotomic incenter	649 (c–a) b2 on Lemoine lattitude line
900 (c–a)(c+a–2b) (∞•~190o)	903 1/(c+a–2b) t ∞•(G—Io)	? (c–a)(c+a–2b)2
812 (c–a)(b2–ca)	335 1/(b2–ca)	? (c–a)(b2–ca)2
? (c+a)(b2–ca) ∞•(Io—tIo)	? (c–a)/(c+a)(b2–ca) --	? (c+a)2(b2–ca)2/(c–a)
(a2–bc+c2–ab)(c–a)	673 1/(a2–bc+c2–ab)
? b(c2+a2–ab–bc)	? (c–a)/b(c2+a2–ab–bc)	- - -
? b(c-a)sbb, ∞•~Go2	? 1/bsbb,	? b2sbbbb,
	From the Gergonne point also the Mittenpunkt
g109 (c–a)sb ∞•~Go	7 1/sb Go	t658 (c–a) sb2
? csc+asa–2bsb ∞•(G—Go)	? (c–a)/(csc+asa–2bsb) -	? (csc+asa–2bsb)2/(c–a)
	From the Symmedian point
523 c2–a2 tS = ∞•~K	83 1/(c+a) tSo	? (c+a)(c2–a2)
524 c2+a2–2b2 ∞•(G—K)	? (c–a)/(c2+a2–2b2) t(∞•(G—K))	? (c2+a2–2b2)2/(c–a)
512 b2(c2-a2) gS = ∞•~tK	310 1/b2(c+a) tpSo	? b4(c+a)(c2–a2) t of Io-Mineur conic perspector
g98 b2SB2-	? (c–a)/b2SB2-	? b4(SB2–)2/(c–a)
	From the Orthocenter also the Circumcenter
? (c2–a2)SB ∞•~H = ∞•~O	27 1/(c+a)SB on Euler	? (c+a)(c2–a2)SB2
30 SBC+SAB–2SCA ∞•(G—K) infinite point on Euler line	? (c–a)/(SBC+SAB–2SCA)	(SBC+SAB–2SCA)2/(c–a)
? b2(c2–a2)SBB	? (c–a)/b2(c2–a2)SBB	? (c+a)b4(c2–a2)SBBBB
	Fissile points
(c–a)(r3 soa ± ∆)	554, 1081 1/(c–a)(r3 soa ± ∆)	(c–a)(r3 soa ± ∆)2
	not so sure what to make of these
? b(c-a)sb(4SBB-c2a2)	? 1/bsb(4SBB-c2a2)	655 (c-a)sb2(4SBB-c2a2)2

Octile points

X554 and X1081 are two of an octile set of points analogous to those discussed here.

The weakened Spieker points, points with roots as coordinates

Since the Spieker points are on the Kiepert hyperbola, the weakened Spieker points wS_o = (:√c+√a:) should lie on the weakened Kiepert hyperbolas and of course they do.

These points are constructed as using François or Paul's construction of the square root point. Although not given in their constructions, the square root point comes in 4 versions and their constructions can easily be extended to cover all versions. The square root of any strong or weak point has 4 real versions.

To construct them, first construct the √I_x, the square roots of the incenter. Take the medial of these points to obtain the weakened Spieker points wS_x.

Tangent Lines

The table shows points on the tangent lines, any one of which could be used to construct the line. Pascal's theorem was used to obtain these as in here.

Tangent line	points on tangent line, defined as concurrence of lines.
through C	gx, BGx—AG, BG—AGx c, gxc•ma—gxb•mc, gxc•mb—gxb•mc, ma, BGx—CG, gxb, CGx—BG, CG—gx•c mb, BGx—gxc•ma, CGx—gx•c
through B	etc
through A	etc
through Gx	a, gxa•mc—gx•c, gxa•mb—gx•b b, gxb•mc—gx•c, gxb•ma—gx•a, CGx—gxa•mc, mb, AGx—gxc•ma ma, CGx—gxb•mc, BGx—gxc•mb c, gxc•mb—gx•b, gxc•ma—gx•a, mc, AGx—gxb•ma, BGx—gxa•mb
through G	a, gxc•ma—gx•c, gxb•ma—gx•b b, gxb•ma—gx•c, gxa•mb—gx•a, AG—gxa•mb, gxc, BG—gxb•ma gxb, CG—gxc•ma, AG—gxa•mc gxa, gxc•mb—CG, gxb•mc—CG, c, gxb•mc—gx•b, gxa•mc—gx•a

Here g_xa = A—G_x, weakened altitude through A.
m_a = median through A.
g_x= weakened Euler line G—G_x.
a = A-edge.
A_G = A-trace of G = midpoint on A-edge.
A_Gx = A-trace of H = foot of A-altitude.

Tangents to Steiner Ellipse and Circumcircle.

The duals of the centers of the weakened Kiepert hyperbolae are tangent to both the circumcircle and the Steiner ellipse. There are four of them. The points of tangency are the perspectors of the four Feuerbach hyperbolae and the four weakened Kiepert hyperbolae.

The medials of these lines are the tangents to the 9 point circle and the inner Steiner ellipse. The points of tangency are the centers of the two hyperbolae, the Feuerbach points and 1086_x.