The isotomic Feuerbach inconics
and their relation to the Feuerbach circumconics and the Feuerbach hyperbolae
The desmic system formed by the intersection of the tFo inconic with its mate. The green points are the 4 conic intersection; the red points their desmic mates, and the blue points is the desmon.
(in progress)
Contents
The isotomic-Feuerbachs and their mates
The iso-Feuerbach inconic has as perspector the isotomic Feuerbach point tFo, which we will call the "isoFeuerbach points." Since it is weak, it acually comes in four versions, one each generated as the isotomic conjugate of one of ~tFx (x = o,a,b,c), the dual lines of the 4 Feuerbach points and with the correxponding Feuerbach point as perspector.
Peter Moses began this investigation by sharing his results on this set of inconics, which contain many famous points. Paul Yiu has an excellent introduction to triangle geometry, including conics.
Looking at all four versions is more pleasing and very interesting.
Notation is here. Generating inconics as the isotomic conjugate of a line is part of the affine theory of triangle conics.
The isoFeuerbach points
and their matesThe original Feuerbach point is the point where the incircle touches the nine point circle. Their isotomics are also an interesting set, not listed in ETC. About them Peter Moses writes
I let my computer have a peek at t11.
t11 seems to be a bit of a wilderness point.
... t11 does not lie on any ETC line, so I can see why it did not appear in early versions of ETC. Even with the current accretion of 3217 points, it is still a bit bereft ...A central point is invariant under permutation of ABC. tFo is central. The triad tFa, tFb, tFc is also central and, it happens, in perspective to ABC at a central point. This point is called the "mate" of tFo and notated by John Conway as (tF)abc and called the "per-isoFeuerbach points, where the "per" is short for "perspective." The mate coordinates are
The two sets of 4 points tFx and per-tFx form a desmic system with ABCDesmon, where Desmon is the strong point . A desmic system is a projective configuration with 12 points and 16 lines, 3 points per line, 4 lines per point. The desmon is the strong point that binds the weak ones together. The Desmon b-coordinate is
The points of a desmic system lie on a cubic whose pivot is the desmon.
A desmic system is best visualized as a projected cube as in the following picture.
The isotomic Feuerbach inconics
and their mates
The development of these four conics closely follows that of the four incircles. The central Feuerbach inconic touches the edges of the central region at three points. The three others touch the central region at three other points that are in perspective at the Feuerbach mate, which is the perspector of an inconic of its own. This is shown in following picture.
Figure: The 4 Feuerbach circumconics are shown in colors.
Properties of the isoFeuerbach-inconics
1. The perspector of the original conic is tFo and its equation is
√x(b–c)2sa + √y(c–a)2sb + √z(a–b)2sc = 0
Since the tFo is always inside the Steiner inellipse, this conic is always an ellipse. Since the other three Fa,b,c are outside, they generate hyperbolas.
As shown above the Nx perspectors imply the Gergonne perspectors Gx, forming a desmic system.
2. The centers are mtFx these points are colinear and on the polar axis.
3. Its contact points at the edges are the Cevian traces of Fx.
Proof: ayz + bzx + cxy = z(ay+bx) + cxy = 0
4. Its duals are the tFx circumconics with centers mtdtFx .
5. asymptotes
6. Its fourth tangent with the Steiner inellipse is the tripolar of the perspector.
7. The axes are parallel to the asymptotes of the Feuerbach hyperbolae. Elsewise the axes are algebraically isolated from most of triangle gometry, as shown in the following picture.
8. Intersection with its mate. Two inconics intersect 4 times in general. One of these points is centrally define. The other three are central as a group and inperspective.
Figure: the tFo conic, its mated conic, and their intersections. There are four intersections, one a central point disignated "o" the other three are a central triad and are perspective at the "per-o" point.
Figure: the desmic system from the intersetion of two inconics. The green points are the four intesections, which form a quartile set. The red points are their mates.
8. Intersection with the incircle is 1317o = : (c+a–2b)2 sbc : and three other points which are perspective at : (c–a)2 sbc :, which does not seem to have a numerical designation. These two points are mates in a desmic system, as shown below. This forms a desmic system similar to the one above.
The dual circumconics
Points
From Peter Moses
I am looking at your conics, but in the mean time I stumbled upon this ...
InEllipse thru X{7, 8, 12, 56, 480, 1259, 1267, 1317, g2446, g2447, t1336, t g {202, 203, 215, 1362, 1672 (COS), 1673 (COS), 1682 (COS), 2007(COS), 2008(COS)}, g t {181}}.
center 3035 = m X(11), perspector t X(11) .. suprisingly tX(11) is not in ETC.
the tangent to Steiner Inellipse touches at sa (b – c)4 (a b – b2 + a c – c2)2 on tX(11) and (b – c)2 (a b – b2 + a c – c2)2 on Steiner
Intersects the Incircle at X(1317)
A point P{p,q,r} on the Circumcircle maps to a4 / ((b - c)2 p2 sa) on the tX(11) inellipse
181 = Apollonius
X(1672) = INSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE)
X(1672) lies on these lines:
1,182 2,1681 8,1680 11,1676 12,1677 55,1343 56,1342 57,1700 181,1683
X(1673) = EXSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE)
X(1673) lies on these lines:
1,182 2,1680 8,1681 11,1677 12,1676 55,1342 56,1343 57,1701 181,1684
x[1672,3] = {a2 (2 b2 c2+a2 SA)±2 a2 b c √S22} an octile point.
The last two references are from ETC. Barycentric coordinates from Jose Cantarella.
Kapetis in Geometry of the Triangle (in Greek) has shown us how to investigate a weak family of conics. His exposition is one of the most beautiful in geometry. He did this for the incircles, but it works for most quartile inconics. It goes like this.
As background, two non-intersecting conics generally have 4 tangent lines. For inconics, three of them are the edges of the triangle, so there is one left, the missing tangent. To find and analyze the missing tangents:
1. Find the tripolars of the perspectors. There are four of them.
2. Find their six intersections, indexed as "ab" for the intersection of the a and b tripolars.
For the Feuerbach conics, these points lie each on one of the six lines of the tIo quadrangle of ABC.
Figure:
3. The "fourth tangents" to these conics are the tripolars of the intersections, six of them indexed in the same way.
4. Each tangent touches 2 conics, the ones whose letters comprise the index, a total of 12 such intersections, 3 per conic forming one triangle per conic.
5. These triangle are each perspective to ABC, which produce four perspectors which are, three at a time, themselves perspective to ABC, forming yet another desmic system. Sometimes the triangles are also perspective to eachother. The central versions of the two perspectors are:
where the last formula is the a-coordinate of the desmon. A beautiful structure within a beautiful structure.
Here are the computations that both verify many statements and give coordinates.
This picture shows the tripolars of the Feuerbachs, their intersections, the "missing tangents," and the triangles, one per conic.