The Mandart inconics
and their relation to the incircles
Contents
Relations between Mandarts and incirclesFeuerbach hyperbola as the organizer of weak conics.
Important parts of this document were contributed by Peter Moses.
The central Mandart inconic has as perspector the Nagel point No. Since is a weak conic, it actually comes in four versions with the four Nagel points Nx are the perspectors where x = o, a, b, c are the four projective directions.
Peter Moses has listed all known points on the Mandart ellipses, 1 of them! – or more precisely, one family of points, the 4 Feuerbach points, one on each conic. Slim pickings indeed! One would think that this would be a boring conic, and it is if you restrict yourself to the central conic. The interesting properties come by considering all four at once.
The best reference on it is Bernard Gibert's paper generalizing the central Mandart conic, beginning with Mandart's original paper. Bernard actually talks very little about the Mandart conic itself in this paper, probably because there seems to be little to say about this solitary conic. Wilson Stothers mention mentions it briefly. Paul Yiu has an excellent introduction to triangle geometry, including conics, although it does not mention the Mandarts.
Notation is here. Generating inconics as the isotomic conjugate of a line is part of the affine theory of triangle conics.
Mated inconics; relation between the Mandarts and the incircles
The incircle touches the edges of the central region at three points in perspective to the Gergonne point. The three excircles touch the central region at three other points perspective at the Nagel point.
The Mandart conics behave oppositely, the central conic has contact points with the Nagel point as perspector, while the external ones have the Gergonne point as perspector. This is shown in following picture.
We term these "mated conics."
For the incircle the original Gergonne point Go is the central perspector, but the other three Ga, Gb, and Gc are together a central object, being in perspective at its mate, the Nagel point. Repeating this process for each region and each Gergonne point, we generate the Gergonne-Nagel desmic system.
For the original Mandart conic, the original Nagel point is the central perspector, but the other three Na, Nb, Nc are together a central object, being in perspective at the Gergonne point. Repeating this process for each region and each Nagel point, we generate the same Nagel-Gergonne desmic system.
I think this is interesting and unusual. But there is more to come.
Figure: The 4 Mandart conics are shown in color. The 4 incircles are shown in light gray. They are both perspective at the same points, the Gergonne and Nagel points. The perspectrices for these points form a desmic system of 12 points and 16 lines, four lines per point and three points per line. Three of the Nagels are off this picture.
The incircle and Mandart central inconic intersect 4 times, one of which is the central point Fo. The other three are central as a group and in perspective at pNo. This forms the desmic system pictured next.
Properties of the Mandart-inconics
1. The perspector of the original Mandart conic is No and its equation is
√x/sa + √y/sb + √z/sc = 0
Since the No is always inside the Steiner inellipse, this conic is always an ellipse. Since the other three Na,b,c are always outside the Steiner inellipse, they generate hyperbolas. This is shown in the next picture.
As shown above the Nx perspectors imply the Gergonne perspectors Gx, forming a desmic system.
2. The centers are the Mittenpunkts Mx = mtNx.
The Mittenpunkt and its extraversions are desmic.
3. Its contact points at the edges are the Cevian traces of Nx.
4. Its duals are the Go circumconics with centers mtdNx.
5. asymptotes
6. Its fourth tangent with the Steiner inellipse is .
7. The axes are parallel to the asymptotes of the corresponding Feuerbach hyperbola.
Elsewise the axes are algebraically isolated from most of triangle geometry, as shown in the following picture.
Figure: Algebraic isolation of the axes. I am just showing this picture of the conics and their axes to show how the axes do not create points as concurrences, illustrating their algebraic isolation. The only significant points they go through are the centers. The same would be true of the asymptotes.
I have no idea why I put the circumcircle is in this 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 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 follow the following steps.
The computations that support this section may be found here.
1. Find the tripolars of the perspectors. There are four of them. Find their six intersections, indexed as "ab" for the intersection of the a and b tripolars.
For the Mandart conics, these points lie each on one of the six bisectors of ABC.
2. The "fourth tangents" to these conics are the tripolars of the intersections, six of them indexed in the same way.
For the Mandart conics, these points lie each on lines harmonically related to the isotomic incenter of ABC.
3. 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.
4. 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 these statements and give the coordinates of these points.
This picture shows the tripolars of the Nagels, their intersections, the "missing tangents," and the triangles, one per conic.
Figure: missing tangents. The tripolars of the Nagel points are shown in bold. Their intersections are indexed by two non-bold letters. Each intersection is on an angle bisector. The tangent lines are light blue each indexed by a pair of bold letters, which indicate the two conics to which the line is tangent. The points of tangency are blue, 3 per conic, and are the vertices of the indicated triangles.
The pro-Gergonnes and pro-Nagels
This picture shows that each triangle is in perspective with ABC at the pGx points, which in turn are perspective in 3's at the pNx points, the pro-Nagel points. These points are centers of similarity of the incircle, circumcircle system. Note: p is the "pro" operation.
Figure: perspectrices and concurrences. The triangle are in perspective at the pro-Gergonne points, which are in turn perspective at the pro-Nagel points.
Here is the desmic system of the proGergonnes and proNagel points, consisting of 12 points, 16 lines, 4 lines per points, 3 points per line expressed as a projective cube.
The missing tangents of the incircles lead to exactly the same desmic system.
The Feuerbach hyperbola as the organizer of weak conics.
A Mineur conic organizes a large group of triangle conics. The Feuerbach hyperbola, in particular, organizes most of the known weak conics, which includes the Mandart conics. This happens because it contains the very special set of points which are themselves conic centers and perspectors.
In our circumstance the four points are the Nagel points No, the Mittenpunkt Mo, the Incenter Io, and the Gergonne point Go. These points include the pairs
No, Go — the line through these points defines the Feuerbach hyperbola.
Inconics (perspector, center)
No, Mo — the Mandart conic, whose axes are parallel to Feuerbach asymptotes.
Go, Io — the incircle
These two conics meet at the Feuerbach center.
Circumconics
Mo, Io — whose axes are parallel to Feuerbach asymptotes
Io, Mo — whose axes are parallel to Feuerbach asymptotes.
These two meet at dFo, which is also on the circumcircle.
Figure: The Feuerbach conic is in red, shown with its most prominent points which themselves produce interesting conics. Go and No are perspectors for the incircle and the Mandart ellipse for which Io and Mo are the centers. Io and Mo are the perspectors for circumconics for which the other is the center. The axes of the ellipses, shown as dashed red lines are all parallel to the Feuerbach asymptotes.
