The Spieker inconics
and their dual circumconics
Contents
The Spieker PointsThe Spieker inconics
Important parts of this document will have been done by Peter Moses.
A weak inconic comes in four versions, indexed by the four projective "directions" o, a, b, c. Other than the incircle, weak inconics have been poorly studied. They are often point poor since the mathematics of inconics implies that few well studied points are on them. The four weak inconics are, as a group, a strong object and hence more likely to have nice relationships, as of course it does.
A way to study a quartile collection of inconics is the method shown to us by Kapetis in his Geometry of the Triangle (in Greek). Reduce the study of inconics to the study of lines, particularly the four tripolar lines of the perspectors, which generate the four conics. In general these four lines will intersect six times, each intersection being tangent to two of the conics.
The Spieker inconics are a special case. Their perspectors are the Spieker points, which are on the Kiepert hyperbola. Their 4 tripolars concur at the Kiepert perspector, which is at infinity, so that the tripolars are parallel and, it happens, perpendicular to the Euler line. The four tripolars meet at only one (infinite) point, instead of the expected six. The tripolar of this point, the dual of the Steiner point, must be tangent to each conic. This is an unusual arrangement, and very interesting.
The isotomic Spieker points are on the G—K line, which is the line that affinely generated the Kiepert hyperbola. Both the Spiekers and the isotomic Spiekers are examples of weak points lying on a single strong object.
The Spiekers are the centers of gravity of the triangle edges in their original and extraverted senses.
Figure: The four Spieker points Sx, which are on the Kiepert hyperbola; the isotomic Spieker points tSx, which are on the G—K line. The black lines are the tripolars of the Spieker points which are hence parallel in the direction perpendicular to the Euler line and parallel to ~K..
The Spieker inconics
Here are the Spieker inconics, all tangent to the line ~S, which goes through the centers of the Kiepert and Jerabek hyperbolas and is tangent to the Steiner inellipse and the nine point circle.
Figure: The 4 Spieker inconics are shown in red, all on the Kiepert hyperbola, shown blue. Their centers are on the G—K line, the symmedian track. The Spieker tripolars are shown as well as the dual of K, to which thy are parallel. Dotted red lines are axes. The green line is ~S or the tripolar of the Kiepert perspector and is tangent to all the Spieker inconics. The Steiner ellipse is shown, the two perspectors inside it producing ellipses, the two outside producing hyperbolas. Dashed green lines are the Kiepert axes.
Properties of the Spieker-inconics
1. The perspector of the original conic is So and its equation is
√(x/(b+c)) + √(y/(c+a)) + √(z/(a+b)) = 0
The others being the extraversions of this one.
Since So is always inside the Steiner inellipse, this conic is always an ellipse. Each of the others could be any type of conic, at least two being hyperbolas and at least one being an ellipse.
2. The centers are the extraversions of mtSo = (: (c+a)(c+a+2b) :) which are on G—K, as the medials of the tSx points.
3. Its contact points at the edges are the Cevian traces of the Sx.
Proof: ayz + bzx + cxy = z(ay+bx) + cxy = 0
4. Its fourth tangent with the Steiner inellipse is the tripolar of the perspector.
5. The axes are ?.
6. Its duals are the tSx circumconics with centers mtdtSx. The centers of the dual circumconics are all on the medial Kiepert hyperbola.
The axes of the dual circumconics are parallel the the Kiepert axes (very interesting).
Figure: The dual Spieker circumconics, all through S, the Steinr point. Their axes are dashed red lines and are all parallel to the Kiepert asymptotes, shown as green dashed lines.
5. Asymptotes for the b-version exist when –3 (a b – a c + b c) +2SW is positive; i.e., when b is greater than the harmonic mean of a and c. The hyperbola degerates when this quantity is zero.
The discriminant for the asymptotes of the o version is -3 (a b + a c + b c)–2SW, which is always negative, so that the o version never has asymptotes. For the b version, it is –3 (a b – a c + b c) +2SW, which can be positive or negative.
Figure: Spieker inconics: 2 hyperbolas, 2 ellipses. Here the discriminant of the b conic is negative, which in this case indicates that it is a hyperbola.
Figure: Spieker inconics: 3 ellipses, 1 hyperbola. Note that disc, the discriminant of the b-conic has changed sign as it has changed form.
