The Feuerbach inconics
and Feuerbach mate inconics
The Feuerbach inconic has as perspector the Feuerbach point Fo. Since it is weak, it acually comes in four versions, one each generated as the isotomic conjugate of one of ~Fx (x = o,a,b,c), the dual lines of the 4 Feuerbach points and with the correxponding Feuerbach point as perspector.
Wilson Stothers mention mentions it briefly. Paul Yiu has an excellent introduction to triangle geometry, including conics, although it does not mention these 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 Feuerbach points themselves
Relation between the Feuerbachs points and the Feuerbach matesThe original Feuerbach point is the point where the incircle touches the nine point circle. By extraversion there are three more where the excircles touch the 9 point circle. These touching points are centers of similarty of the incircle-nine point circle system.
Figure: The incircles touching the 9 point circle at the four Feuerbach points.
A central point is invariant under permutation of A, B, C. Fo is central. The triad Fa, Fb, Fc is also central and, it happens, in perspective to ABC at the Feuerbach mate Eo, also written Fabc, as in the pictures below. These points are the 8 centers of similarity of the incircle, nine point circle system.
Figure: the four guiding lines of the Feuerbach desmic system, a set of four weak lines going through the weak points Ix, Fx, Ex concurring at a strong point, N. Sometimes I call thse lines the "spines" of the desmic system.
The two sets of 4 points Fx and Ex form a desmic system with ABCN. 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 points of a desmic system lie on a cubic whose pivot is the desmon.
A desmic system is best visualized as a projected cube — in this case ABCN and Fx are opposite vertices of the cube, whose edges concur at the Ex.
Figure: The Feuerbach desmic system.
The Feuerbach points are also the center of the Feuerbach hyperbolas.
The 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 Eo, which is the perspector of an inconic of its own. This is shown in following picture.
Figure: The four Feuerbach ellipses are shown in blue. The green inconic is the mated conic of the central Feuerbach one. It touches the triangle edges at the same spots as the Fa, Fb and Fc inconics. The axes of the Feuerbachs are the dashed red lines. One can see the Algebraic isolation of the axes. The only significant points they go through are the centers.
Figure: The 4 Feuerbach conics are shown in blue, their mates in green.
Properties of the Feuerbach-inconics
1. The perspector of the original conic is Fo = (: (c–a)2sb : ) and its equation is
√x/(b–c)2sa + √y/(c–a)2sb + √z/(a–b)2sc = 0
Since the Fo is always inside the Steiner inellipse, this conic is always an ellipse. Since the other three Fa,b,c are outside, they generate hyperbolas.
The Fx perspectors imply the Ex, forming a desmic system.
2. The centers are mtFx = (: (c–a)2sb (abc – 4sabc – b2sb) : ) 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 tFo circumconics with centers mtdtFx .
5. asymptotes -- none
6. Its fourth tangent with the Steiner inellipse is .
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.
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 bisectors of ABC.
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.
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.
