The Eulerian Hyperbola,
inParabola, CircumParabola
and associated structures

(in progress)

Contents Eulerian and dual conics Properties Points Peter Moses has made important contributions to this document.

A listing of the notation is found here.

Eulerian hyperbola, Eulerian inparabola, and circumparabola

I have developed an organizational scheme for circumconics. The scheme from strong conics is shown in the first figure below and implies that there should be two important conics derived from lines through G; these being the Kiepert hyperbola, being the conjugate of G—K, and a new hyperbola from the conjugate of the Euler line G—H. This last one I have not found in the standard references, so I am calling it the Eulerian hyperbola. We will present this hyperbola as an immediate deduction of the affine theory of the triangle.

The Eulerian hyperbola completes a basic symmetry between Brocardian structure (from the G—K line) and Eulerian structure (from the G—H line).

Notation is here.

Consider the orthocenter H. There are two affine invariant lines associated with it: G—H and ~H, the dual of H. These lines are invariant in that, after an affine transformation, H becomes P, say, G—H becomes G—P and ~H 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 ~H produces the H circumconic, whose perspector is H, and the line G—H produces the Eulerian hyperbola, whose perspector is ∞•~H, which is at infinity. The schema below, part of the much larger schema of strong conics, shows many relationships, and shows that the strong conics are related to one master strong conic, the Jerabek hyperbola.

Figure: the cycle of points K —d— D —t— H —m— O, all on the Jerabek hyperbola, gives a picture of many conic relationships. One of these is the Euler line O — G — H, which generates the Eulerian hyperbola and parabolas discussed here.

Figure: The Jerabek (red), Kiepert (blue), and Eulerian (green) hyperbolas and supporting lines.

Almost all of the properties listed below are shared by any other circumconic with perspector at infinity (i.e., derived from a line through G). The exceptions are noted in the item listing and are two: the fact that ~D is perpendicular to the Euler line and the ~H is perpendicular to H—K.

1. Definition: The Eulerian Hyperbola is the isotomic conjugate of the line G—H. Its perspector is ∞•~H = (: (c²–a²)S_B :), the dual of G—H and the infinite point on ~H. The equation of the Eulerian hyperbola is

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

The Eulerian hyperbola is also the isogonal conjugate of K—pH, where pH is Gob's center. The affine approach used here derives a great number of properties rapidly, with little effort, as well as puts this hyperbola in a larger context of triangle conics.

2. Asymptotes: The line G—H 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 endpoints of the asymptotes are the conjugates of the intersections of G—H with the Steiner ellipse, which are

The intersections of G—H with the Steiner ellipse are ( : b²S_B–2S_CA ± Z : ) where Z = √(S⁴–3S_WABC). Their conjugates are the endpoints of the asymptotes. The equations of the asymptotes are ( : (c²–a²)S_B (b²S_B–2S_CA ± Z)² : ) which can be shown to be conjugate.

3. Axes:

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

Figure: The Eulerian hyperbola. This picture shows the hyperbola in blue with its center, foci (green, red text), asymptotes, together with significant points, lines and circles. The two Steiner ellipses are shown in cyan. 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 in red. Q_1,2 are the intersections of the Euler line with the circumcircle. See point notations linked below. The structure of lines from points on the conic and the Steiner ellipse is shown.

5. Center: The center for a circumconic is mtd of the perspector and, for the Eulerian hyperbola, it is (: (c²–a²)² S_B² :) which is on the Steiner inellipse. The center is also on the nine point circle since the centers of all rectangular hyperbolae are there.

The dual of the Eulerian center is tangent to the Steiner ellipse at the conjugate of the perspector t ∞•~H .

Associated lines: Just as the Euler line gives us a rigidly defined structure of points, so does the Steiner ellipse. The line that connect these points also go through important points on this conic. This is shown in the figure above.

6. The Eulerian parabola: The dual of the Eulerian hyperbola is an inparabola (as is the dual of any hyperbola with perspector at infinity), the Eulerian parabola, whose perspector is the Steiner point t∞_H, the conjugate of the hyperbola perspector ∞•~H. Its focus g∞_H is on the circumcircle. The line from the perspector to the focus of an inparabola goes through S. The axis is through the focus parallel to ~H, i.e., perpendicular to the HK line and parallel to the polar axis. The directrix is perpendicular to ~H through H, which makes it the KH line. The simpson line of the focus is parallel to the directrix H—K through the vertex.

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

Since a conic and its dual never intersect, the Eulerian 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 Eulerian hyperbola to the Eulerian parabola this is y -> : (c² – a²)S_B/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 mt∞•~H on the Steiner inellipse, and center ∞•~H, at infinity. Like the Eulerian Parabola, its axis is also in the direction of ~H, with endpoint ∞•~H.

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

The switched circumparabola is the conjugate of the dual of the Eulerian 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 Eulerian hyperbola with the circumcircle is ng∞_H the isotomic of the intersection of ~H and G—H. Its fourth intersection with the Steiner ellipse is te_∞, the reflection of ∞_H in G and the isotomic of the infinite point on e = G—H and the dual of the simpson line of the parabola focus.

Figure: The Eulerian circumhyperbola; its dual, the Eulerian 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 Eulerian parabola. The axis of the Eulerian parabola is parallel to ~H and perpendicular to the HK line, which is its directrix. 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 in red.

10. Tangents at important points: G—H is tangent to the Eulerian hyperbola at G. A line parallel to G—H is tangent at t∞•e.

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

Tangents at the vertices are parallel to the minor axis.

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.

Points

Peter Moses contributed this list of points on this conic. I add the coloring: blue means strong; black, super-strong; red, weak. Light blue are those I do not think are very interesting.

ABCGD. Isotomic of Euler line,
Isogonal line K X(25) Gob
ETC points on hyperbola
{2, 69 D, 95 (1/(Q²-b⁴-c²a²), 253, 264 (tO), 287 S_B/S_B^2–, 305 t Gob, 306 (c+a) S_B, 307 (c+a)S_B/s_b, 328, 1441 tH_o=(c+a)/bs_b, 1494, 1799 S_B/(c²+a²), 1972 , 2373, 2419, g159 (g of M_o of Tangential) , g161, g206 gsO, g1194, g1660, g1915, g2203}
Center = mX(648) {{3, 67 1/(S4_B-c²a²)}/\{115 mS, 127} , {122,125 J} ,...},
perspector X(525) at infinity.
4th meet with CC = X(2373) = ng∞_H
4th meet with Steiner Circumellipse = X(1494) = te_∞.
 
A point P{p,q,r} on the CC is mapped to ABCGD by p (b² - c²) S_A /a² ::
A point P{p,q,r} on the Steiner circumellipse is mapped to ABCGD by p (b² – c²) S_A ::
A point P{p,q,r} on L.inf is mapped to ABCGD by (b² - c²) S_A / p ::