Inconics
The Spieker inconics, a quartile set, all tangent to S, the Steiner axis, dual of the Steiner point (green line). Dashed red lines are axes. Heavy lines are the duals of Spieker points. |
Contents. Strong MacBeath inconic (and dual) Weak |
The special nature and termininology of inconics.
1. Inconics touch the three sides of the reference triangle ABC, one point per side. If the point on the A-side of the triangle is connected to A and correspondingly for B and C, these 3 lines concur at the perspector of the inconic. The coordinates of the perspector ( p : q : r) determine the equation of the conic as √(x/p) + √(y/q) + √(z/r) = 0, where all square roots can take both signs.
It takes 5 points to determine the general conic. If the conic is inscribe, it only takes one, the perspector.
2. The center of an inconic is mtP, where m is the medial operation and t is the isotomic conjugate. The perspector and center uniquely determine eachother, so that knowledge of the center also determines the conic. The perspector is td of the center, where d is the dilated or antimedial operation.
3. The foci are isogonally conjugate. This means that knowledge of one focus determines the other. The center is the midpoint of the two foci, hence knowing one focus determines the conic.
To create a conic from one focus, find the isogonal conjugate from that focus, this being the second focus; take the midpoint of the two foci, this is the center; find the perspector from the center.
The isogonally conjugate foci determine a circumconic known as a Mineur conic whose asymptotes intersect the inconic at its vertices.
4. If the perspctor is weak, there usually a corresponding, or "mated," conic. The four intersections of these two form a desmic system witb respect to ABC. A weak inconic comes in four versions.
5. The intersections of inconics form a desmic system.
Peter Moses has listed the ETC points on many inconics. There are few. Lemoine has 1, MacBeath had 2 until the Hyacinthos groupmade a concerted effort to find some, the Mandart has 1, the incircle only a few (I suspect created just so it would have some), the orthic has 1, the D-inconic has 2, Brocard 5. There are almost no inconics with more than a few.
The ones with the most seem to be unstudied: perpector tFo, the isotomic Feuerbach points (not in ETC) with 9, and the one with perspector X59, the isogonal of the Feuerbach center, with 15.
But inconics are line-conics, not point conics. They are enveloped by known lines, which can lead to new points in the form of points of tangency (although few will be in ETC).
These inconics become point (and line) rich if you consider the pair if mated inconics. In this I will use the example of the mated pair of the incircles (perspectors the Gergonnes Gx) and the Mandarts (perspectors the Nagels Nx). I put these inconics in the plural because they occur as sets of 4, called weak conics.
It is easy to study the central versions of these conics only, which is likely to lead to the conclusion that the conic is uninteresting. With the incircles, we have learned that they are much more interesting if one includes them all. Together they form a strong object, not a weak one, and will thereby more likely to have interesting properties.
The original incircle touches the triangle at the traces of Go, the central Gergonne point. The central Mandart touches at the traces of No.
But the other three conics in each quartile set together form a central object. The other 3 incircles (aka, the excircles) touch the central region of the trinagle at the traces of No (same as the central Mandart). The 3 ex-Mandarts touch the central region in the traces of Go.
This is mated behavior. Each set makes use of points from the other set. The terminology and the importance of desmic sets is due to John Conway.
What we know from the incenters (due to Kapetis) is that the 6 4th tangent lines lead to a new desmic set using pGo = X55 and pNo = X56.
It turns out that the 4th tangents to the mated Mandart inconics leads to the same set.
These desmic structures lead to many central points.
In addition two mated conics meet four times, which are the 4 Feuerbach points, which are desmic with X41, whose significance I do not know.
So inconics have lots of interesting structure, if you consider them all, and if you look at the lines on them rather than the points on them.
Mittenpunkt and per-Mittenpunkt
inconics
with interlocking desmic systems.
Spieker inconics
which are all tangent to the ~S line which goes through the Kiepert and Jerabek centers.
Feuerbach and per-Feuerbach inconics
isotomic Feuerbach inconics and Feuerbach circumconics
MacBeath inconic, O circumconic, D and H inconics
for which G and K are foci.