The Mittenpunkt inconics
and their Mineur circumconic, and their focal conic
(in progress)
Contents
Desmic system terminologyFeuerbach hyperbola as the organizer of weak conics.
Important parts of this document were contributed by Peter Moses.
The weak points known as the Mittenpunkt and its extraversions have many wonderful properties. These properties lead to equally wonderful properties in their inconics.
The points and their mates, the isogonal Mittenpunkt points, form a classically beautiful Desmic system (shown below), with the centroid as desmon.
Notation: o,a,b,c index the four projective "directions." Coordinates are always barycentric and usually shown as the middle coordinate.
Many triangle objects exist in configurations. Some are three-fold, such as the three vertices A, B, C or the medians. The three coordinates that we use for a point are a natural reflection of this. But there are also two-fold objects such as the isodynamics points, and four-fold objects, such as the incenters and incircles.
It is tempting to think of the three fold symmetry that permutes the triangle vertices as the fundamental one, but it is not. The lines of the triangle divide the projective plane into four regions. The permutations of these regions is a very powerful group which includes the permutations of ABC. The four-fold triangle plane is more fundamental than the three-fold triangle.
There are four Mittenpunkt points, one, Mo = ( : b sb : ), being central. The other three Ma, Mb, Mc also form a central object. When this central object is in perspective at what we call the Mittenpunkt mate Mabc = gMo = ( : b/sb : ). These eight points are in two groups of four, auspiciously arranged. The 12 lines from ABC to either group of 4 goes through the other group, making 11 points and 12 lines, 3 points per line, 3 lines per point. This is not quite a complete projective configuration, which is obtained by connecting corresponding points for four more lines, which in this case concur at G. Finally and completely there are 12 points, 16 lines, four lines per point, three points per line.
This can be arranged into a projected cube as shown in the figure. This figure is named by John Conway a "desmic system," with the strong central point, G in this case, called the desmon. The 8 vertices are from two quartile sets, in this case the Mx and gMx with opposite vertices of the "cube" being corresponding points. The 12 edgelines concur in fours at ABCG, completing the desmic system.
This picture below of the two inconics shows how the mitten mate gMo is created from the mitten extraversions Mc, Mb, Mc, and also that the perspectors and centers are collinear with the symmedian point.
In general two inconics meet at 4 points, one central, the other three central as a group and perspective to ABC. In our case the central point is (: b (c–a)2 sb :) and the perspector is 41o = pMo = (: b3 sb :). These four points are part of a new desmic system. Note that 41o refers the the central version of the point X41 from ETC.
Figure: This picture of the two inconics shows how the mitten mate Mabc = gMo is created from the Mitten extraversion points Ma, Mb, Mc (dashed red lines), and that the perspectors and centers are collinear with the symmedian point. The two central points from the conic intersections are shown, one being pMo = 41o.
The Mo Mineur circumconic
Well that was interesting enough, but there are new, unique elements to this situation. The Mittenpunkt and the isogonal Mittenpunkt are conjugate. This has two consequences. First they can each be the focus of a new inconic, which we will call the focal inconic. Also the line joining these two points is a Mineur line, whose conjugate is a Mineur conic. These two points and their supplements are on the conic. The Mittenpunkt supplement is K the symmedian point, while the other is pNo, a center of symmetry of the circumcircle and the incircle. These are two very interesting points.The Mineur conic is a hyperbola that follows the sweep pattern through the triangle.
The asymptotes of the Mitten-Mineur hyperbola meets the Mitten focal inconic at its vertices.
Figure: The blue ellipses are the inconics of Mo and gMo. Since these points are conjugate, they are the foci of an inconic shown in red. In addition the line between them generates a Mineur conic, shown as the blue hyperbola. The asymptotes of the Mineur hyperbola intersect the focal conic at its vertices. The dual inconic, perspector tP, center mP, to the hyperbola is also shown in light blue.
The tripolars of the four conspicuous points on the hyperbola meet at the hyperbola perspector.
Points
Information on this Mineur conic from Peter Moses. As always, a Mineur conic contains lots of points. As usual I color the points red if they are weak and blue if strong and cyan if octile. Since there is one strong point on this conic (there cannot be more), all versions of this conic go through K. X(9) is the Mittenpunkt.
gX(9) = X(57)
The {X(9), gX(9)} line is the line through ...
{2, 7, 9 Mo, 57 gMo, 63 Wo, 142,144,226,307,329,527,553,579,672,894, 908,1025,1400,1423,1445,1447,1652,1653,1708,1741,1944,2002,2094,2285,2406}
The isogonal of this is a circumhyperbola, perspector X(663), cyclic sum a2sa(b – c) y z = 0 through
{6, 9 Mo, 19 gWo, 55, 57gMo, 284 a2 sa/(b+c), 333 sa/(b+c), 673 mIo2-=1/(b2–+c2–), 893 pIo2-=a2/a2+, 909 a2/(-4sabc+aSA ), 1024 a2sa(b–c)/a2–, 1174,1436 a2/(–2sabc+aSA ), 1751,1945,2160 b / (a2–b2+a c+c2), 2161 b / (a2–b2–a c+c2), 2164,2195 a2sa/a2–, 2258,2259,2291,2299 a2 sa/(b+c)SA, 2316 a2 sa/(b+c–2a), 2319 a2 sa/(ab+ca–bc), 2337,2339,2343,2364 a2 sa/(2b+2c–a), 2432, 2590,2591} (last pair have √(abc(abc-8sabc))
center a2 sa (b - c)2 (abc –(a–b)(c–a)(b+c) + 2a2sa).
Call the conic CC663
4th intersection with circumcircle, X(2291) (tangent line {1055,2078})
4th intersection with Steiner, {{55,190}/\{99,284}, ..}
4th intersection with Kiepert, X(1751)
4th intersection with Feuerbach, X(9) (tangent line {8,9, ..})
4th intersection with Jerabek, X(6) (tangent line {6, 41,..})
4th intersection with (isotomic of Io — tIo ), X(19) (tangent line {19,208})
Point {p,q,r} on circumcircle, sa (b - c) p :: is on CC663
example ... X(98) -> (b – c) sa / (SA2– ) :: = {{6,523,879,1316}/\{19,798}/\{284,522}/\{333,652}/\{650,893} .. }
Point {p,q,r} on Steiner, a2 (b – c) sa p :: is on CC663
example . X(671) -> a2 (b – c) sa/ (2 a2 – b2 – c2) :: = {{6,512}/\{284,663}/\{333,522}/\{673,897}/\{909,923}, ..}
Point {p,q,r} at infinity, a2 (b – c) sa / p :: is on CC663
Figure. The Mittenpunk Mineur conic is in red, the Feuerbach conic, also a Mineur conic, in blue. Their fourth intersection is Mo, lines from which go through corresponding points in each conic. The red ellipse is has the Mo and gMo as its foci.
Figure: The 4 Mitten inconics are shown in blue. The centers mtMx are show. The axes are dashed and red.
Properties of the Mitten-inconics
The properties of inconics are not as interesting as those of circumconics.
1. The perspector of the original conic is Mittenpunkt Mo and its equation is
√x/(asa) + √y/(bsb) + √z/(csc) = 0
and extraversions.
Since the Mx are always inside the Steiner inellipse, these conics are always ellipses.
As shown above the Mx perspectors imply their mates, it isogonal Mittenpunkt's gMx, forming a desmic system.
2. The centers are mtMx, the "o" version of which = ( : b sb (b sb + 2 sca) : )
3. Its contact points with the edges are the Cevian traces of Mx.
Proof: ayz + bzx + cxy = z(ay+bx) + cxy = 0
4. Its duals are the tMx circumconics with centers .
5. No asymptotes.
6. Its fourth tangent with the Steiner inellipse is ....
7. The axes are ?.
The axes are algebraically isolated from most of triangle geometry, as shown in the following picture.
The four isogonal Mitten inconics
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.
Figure: The centers and perspectors of both conics are collinear. with K.
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.
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 Mitten conics, these points lie each on one of the six Cevian lines of 19o = gWo
2. The "fourth tangents" to these conics are the tripolars of the intersections, six of them indexed in the same way.
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. The two perspectors are.
The Desmon is 1/SB2. A beautiful structure within a beautiful structure.
Here are the computations that verify many of the statements and give the coordinates of many points and lines.
This picture shows the tripolars of the Nagels, their intersections, the "missing tangents," and the triangles, one per conic.
Figure: the missing tangents. The tripolars of the Mittenpunkt points are shown in bold. Their intersections are indexed by two non-bold letters. The tangent lines are light, each indexed by a pair of letters, which indicate the two conics to which the line is tangent.
This picture shows the contact points are in perspective to ABC at a new central point
Figure: "oc" is the tangent to the "o" and "c" versions of the inonics. The points of tangency of oa, ob, oc on the Mo-inconic are shown as a triangle and are perpective to ABC. The red point is the perspector, the first of four.
Figure: perspectrices and concurrences. Each of the four conics has a triangle analogous to the one above. The triangles are in perspective at 4 new points.
Figure: The perspectors (red points) are in perspective (green points).
The desmic system. The missing tangents of both the Mittens and the Mitten mates' inconics lead the same desmic system. This is the same picture as the last, rendered to make the desmic system easily visible.
