MacBeath inconic
and the O-circumconics and H, D inconics
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This movie shows the construction of the MacBeath conic as the envelope of the perpendicular bisectors of PH, where P is on the circumcircle. The red circle is the locus of the midpoints of PH and is the nine point circle. |
Contents:
Important parts of this document were contributed by Peter Moses. |
Today's special is the MacBeath conic, which has a nice method of generation. Choose a point P on the circumcircle; the midpoint M of PH lies on the 9 Point Circle, while the perpendicular bisector of PH is tangent (at T) to the MacBeath conic. This is shown in the animation above, the final result is in the following picture. The circumcircle is the director circle and the 9 point circle is the auxiliary (or principle) circle.
MathSource says that MacBeath announced this conic as the inconic with the greatest area, which I doubt.
We will analyze this conic by showing properties that come from the locus definition and as a derived conic from the Jerabek hyperbola.
The MacBeath conic as envelope of the perpendicular bisectors of PH, where P is on the circumcircle.
A year or so ago Hyacinthos was involved in a discussion of points on the MacBeath conic. In older versions of ECT there were few listed points on this conic. Points were found by mapping points on the circumcircle or other geometric structure to a MacBeath conic. Peter Moses has given us a list of formulas that will implement this. But of course one wonders if this purely mathematical way of finding points gives points of any geometric relevance. The argument in its favor is that natural points on one conic will lead to natural points on another, but of course this begins as conjecture, not fact.
I have been very concerned with this problem of creating reasonable points on geometric objects. To do this I have taken the simplest curve, the line at infinity. As integers can be developed from special integers call the primes, so can central points be developed from special central points on the line at infinity. Initially I thought this was just an interesting game, but I've found that the wisdom involved in these points and their mappings to any conic is in fact almost the same wisdom as shown in the history of triangle geometry. I call this mapping from points at infinity to other objects the natural mapping, and it gives results equivalent to Peter and Wilson's mappings from the circumcircle.
MathSource has a page on this inconic and the AbraCadaBri site (in French) discusses this method of generating a conic. Wilson Stothers mention mentions the MacBeath conic briefly. Paul Yiu has an excellent introduction to triangle geometry, including conics. He mentions this conic, but not using the name of MacBeath. I learned about it from gonzo geometer Gene Olmstead.
Figure: The MacBeath conic is red, Jerabek hyperbola is blue, and the H and D inconics are black. Points on the MacBeath conic are shown, blue for strong, red for weak, and green for fissile. The number next to a point is its ETC index. A designation such as 2968b indicates it is the "b" member of a quartile set. The affine parallelogram of H is shown as green.
So let's look at the points on the circumcircle and the MacBeath inconic. Using the locus definition, each point on the circumcircle generates a point on the MacBeath conic. We will begin with important points on the circumcircle and look at the points they generate. The most important points on the circumcircle are the Steiner and Tarry points, the Focus point, and points that correspond to the directions of strong lines. For the P = T, the Tarry point, the midpoint is mS, the center of the Kiepert hyperbola, and the tangential point is X339, the retro-Jerabek point. I learned this from Nikolaos Dergiades. A list of these correspondences given below.
The two intersections of the line P—H on the circumcircle generate antipodal points on the MacBeath ellipse.
In the following pictures the MacBeath conic is red; the circumcircle and 9 point circles are shown.
 The Tarry point on the circumcircle generates rJ = X339 on the MacBeath inconic. |
 The Steiner point on the circumcircle generates X2974 on the MacBeath inconic. |
 The second intersection of T—H on the circumcircle generates X2967, the antipode of X339 on the MacBeath inconic. |
The second intersection of S—H on the circumcircle generates X2971, the antipode of X2974 on the MacBeath inconic. |
But how does the locus generation of a point correspond to the natural mapping? Very nicely it turns out. Under the natural mapping the Tarry point goes to the opposite point on the conic. The point on the circumcircle that generates this opposite point is found by moving from the Tarry point through the orthocentre to the other intersection with the conic. This procedure is generally true. This is shown in the above pictures.
The natural mapping produces the antipodal point on the conic from the locus definition. A table showing the natural mapping and the points produced is below.
The following table shows the relation of the point P on the circumcircle, the P—H midpoint M, and the tangential point T where the PM perpendicular bisector touches the conic. In the last two columns are the antipodal point, which results from the natural mapping, and its natural point (the point of tangency of its dual) on the MacBeath circumconic. The dual of the points in the last two columns is tangent to the other conic at the indicated point. The numbers of the table are the index of the point in ETC.
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P
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M
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Tangency
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antipode
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on MacBeath circumconic
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T, the Tarry point
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mS
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339
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2967
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287
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e∞, the Euler intersection with circumcircle
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?
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1312,3
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1313,2
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?
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2968
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651
isotomic Feuerbach perspector |
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X101
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2973
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1331
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X100
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2969
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1332
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F
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2970
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S
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?
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2971
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2974
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ge∞
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J
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2972
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?
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2972
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648
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X109
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1813
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So these points on the conic work out nicely as ETC points. Some of the points are weak. Weak points come in four versions. The intersection of two inscribed conics happens four times. They intersection of strong inconics forms a quartile set of weak points. Wilson has shown us that if a quartile set is on a circumconic, it cannot be Desmic. Interestingly if a quartile set is on an inconic it must be Desmic. There are two interesting things to ponder here. First is the fact that intersecting inconics produce pairs of quartile points in a Desmic system. Second is that we can look for the origin of quartile points on an inconic as the intersection with another inconic. This is a very successful strategy for the MacBeath conic.
But which inconics? Here the Jerabek hyperbola comes to our aid. The MacBeath conic has two very famous points as foci, the circumcenter and the orthocentre. The foci of an inconic are always isogonal conjugates. To conjugate points also define a special conic that I call the Mineur conic, and a conic of high symmetry. Each Mineur conic has a special set of inconics and circumconics, to which we turn our attention.
The orthic inconic (perspector H, center K) and the D-inconic (D and O) both have their perspectors and centers on the Jerabek hyperbola. The orthic inconic intersects the MacBeath inconic at X2968, and the other one at X2967, both weak points.
Below is a table showing the natural points and their colleagues on the MacBeath inconic.
The major axis of the MacBeath conic is the Euler line. The major axis vertices of the McBeath conic are the intersections of the Jerabek asymptote with the Euler line. This is a general property of Mineur conics and vocal inconics. The 9 Point Circle also goes through these points.
The MacBeath inconic properties
1. The perspector is P = tO, the isotomic circumcenter. It is enveloped by the dual of points on ~O the tripolar of the perspector. The inconic equation is
√(a2SA x) + √(b2SB x) +√(c2SC x) = 0
where the square roots are understood to be both + and –.
The position of the perspector compared to the Steiner ellipse determines the nature of the conic: inside gives an ellipse, on gives a parabola, outside gives an hyperbola. The MacBeath conic is an hyperbola when the triangle is obtuse.
2. Its center is mtP = N, the nine point center. If the perspector is on the Steiner inellipse, then the center is on the medial triangle. So we can characterize the type of conic by the position of the center to the medial triangle: inside implies ellipse, on implies parabola, and outside implies hyperbola. The nine point center is on the medial triangle when the triangle is right.
3. The foci are two well known points O and H.
3. The contact points with the triangle edges are the traces of tO.
4. Axes : The axes are the Euler line and its perpendicular at the center N.
Here are the conic variables for the MacBeath conic. y = R/ OG, which Conway calls the Eulerian parameter. The MacBeath conic variables were computed by John Conway.
The MacBeath inconic
= R/2 √( 1 – 9/y2)
MacBeath in relation to the Jerabek hyperbola.
The Jerabek hyperbola
The H and D inconics
The Jerabek hyperbola contains the points H, K, O, D of which H, O are isogonally conjugate and H, D are isotomically conjugate. H and O are the two foci for the MacBeath inconic. A circumconic that contains conjugate points has very special properties. In particular it contains the center and perspector of many conics.
H and K are the perspector and center of the orthic inconic. These points are on the Jerabek hyperbola.
D and O are the perspector and center of the D-inconic. These points are on Jerabek.
The foci of the MacBeath ellipse are on the Jerabek hyperbola.
As we would expect there are many connections between these conics.
1. The asymptotes of the Jerabek hyperbola go through X1312 and X1313, the vertices of the MacBeath inconic.
2. The axes of the H and D inconics are parallel to those of Jerabek.
3. The orthic (H) inconic meets MacBeath at the four versions of X2969.
4. The D-inconic meets MacBeath at the four versions of X2968.
5. The H and D inconics go through the Jerabek center.
6. The MacBeath circumconic (center K, perspector O) has axes parallel to the Jerabek asymptotes.
Figure: The Jerabek hyperbola is red. The MacBeath inconic is blue. The D and H inconics are black. Weak points are red, strong, blue, and fissile green. The asymptotes of the hyperbola go through 1312, 1313, the major vertices of the MacBeath ellipse. Blue dashed lines are asymptotes of the Jerabek hyperbola. Red ones are axes of the O circumconic and the H and D inconics, all of which have axes parallel to the Jerabek asymptotes. The affine parallelogram of H and D is yellow.
The MacBeath inconic has a dual, the MacBeath circumconic, which has perspector O and center K.
From Peter Moses we have the points on 4 conics of interest
Note: blue points are strong (1 version), red are weak (4 versions), and green are fissile (2 versions). Points rendered lightly are being ignored. Coordinates given are the y-barycentric coordinate.
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MacBeath circumconic, center 6, perspector 3 |
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D inconic 69, center X(3), |
The following chart organizes both points and lines. Each entry has the y coordinate (in black) of some line or point. The ETC number is red and the Geometry interpretation, if known, below. Each entry can be doubly interpreted, as a point for some conic or as a line for the dual conic. The top row in black and yellow is for points: the second row in yellow and black is the dual of the first and is for lines. For example ( : (c–a)2/SB : ) is the y coordinate of a point on the MacBeath inconic (perspector tO) and [ : (c–a)2/SB : ] is the y-coordinate of the line tangent to the MacBeath circumconic (perspector O).
The colors divide the chart into sections whose structure can be said to originate from the named points, as explained here.
Just for fun the H and D inconics, which include few ETC points, are included.
A red mark to the right of a point means that it is not listed in ETC.
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line at infinity
(point) |
circumcircle
(point) |
MacBeath inconic
(point) |
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MacBeath circumconic
(point) |
H inconic
(point) |
D inconic
(point) |
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Steiner ellipse
(line) |
tK inellipse
(line) |
MacBeath circumconic
(line) |
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MacBeath inconic
(line) |
D circumconic
(line) |
H circumconic
(line) |
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y
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b2/y
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y2/b2SB
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b2SB/y
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y2/SB
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y2SB
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From the Incenter
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514 c–a
twS =∞•~Io |
101 b2/(c-a)
pwS |
2973 (c–a)2/b2SB
- |
1131 b2SB/(c–a)
- |
1269 (c–a)2/SB
- |
1565 (c–a)2SB
- |
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519 c+a–2b
∞•(G—Io) |
106 b2/(c+a–2b)
- |
? (c+a–2b)2/b2SB
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? b2SB/(c+a–2b)
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? (c+a–2b)2/SB
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? (c+a–2b)2SB
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513 b(c–a)
∞•~tIo |
100 b/(c-a)
wF |
2969 (c–a)2/SB
intersection with H inconic |
1332 b SB/(c–a)
- |
? b2(c–a)2/SB
- |
? b2(c–a)2 SB
- |
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900 (c–a)(c+a–2b)
(∞•~190o) |
903 1/(c+a–2b)
t ∞•(G—Io) |
? (c–a)2(c+a–2b)2/b2SB
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? b2SB/(c–a)(c+a–2b)
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? (c–a)2(c+a–2b)2/SB
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? (c–a)2(c+a–2b)2SB
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812 (c–a)(b2–ca)
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? b2/(c-a)(b2–ca)
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? (c–a)2(b2–ca)2/b2SB
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? b2SB/(c–a)(b2–ca)
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? (c–a)2(b2–ca)2/SB
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? (c–a)2(b2–ca)2SB
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| 518 b(c2–+a2–) |
? b/(c2–+a2–)
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? (c2–+a2–)2/SB
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1814 bSB/(c2–+a2–)
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? b2(c2–+a2–)2/SB
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? b2SB(c2–+a2–)2
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? b (c–a) sb SB
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b / (c–a) sb SB
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2968 (c–a)2 sb2 SB
intersection with D inconic |
651 b/(c–a) sb
isogonal Feuerbach perspector |
? b2(c–a)2 sb2 SB
- |
? b2(c–a)2 sb2 SB3
- |
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From the Gergonne point
also the Mittenpunkt |
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g109 (c–a)sb
∞•~Go |
109 b2/(c-a)sb
- |
? (c–a)2 sb2/b2SB
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1813 b2SB/(c–a) sb
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? (c–a)2 sb2/SB
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2968 (c–a)2 sb2SB
intersection with D inconic |
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From the Symmedian point
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523 c2–a2
tS = ∞•~K |
110 b2/(c2-a2)
F |
2970 (c2–a2)2/b2SB
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? b2SB/(c2–a2)
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2969 (c2–a2)2/SB
intersection with H inconic |
125 (c2–a2)2SB
J |
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? (c2–a2)(b4–c2a2)
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(c2–a2)2(b4–c2a2)2/b2SB
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b2SB/(c2–a2)(b4–c2a2)
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(c2–a2)2(b4–c2a2)2/SB
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(c2–a2)2(b4–c2a2)2 SB
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512 b2(c2-a2)
gS = ∞•~tK |
99 1/(c2-a2)
S |
2971 b2(c2–a2)2/SB
- |
? SB/(c2–a2)
- |
? b4(c2–a2)2/SB
- |
? b2(c2–a2)2SB
- |
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g98 b2SB2-
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98 1/SB2-
T |
2967 b2(SB2–)2/SB
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287 SB/SB2–
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? b4(SB2–)2/SB
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? b4(SB2–)2SB
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From the Orthocenter
also the Circumcenter |
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525 (c2–a2)SB
∞•~H = ∞•~O |
112 b2/(c2–a2)SB
- |
339 (c2–a2)2SB/b2
rJ |
110 b2/(c2–a2)
F |
125 (c2–a2)2SB
J |
? (c2–a2)2SB3
- |
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30 SBC+SAB–2SCA
∞•(G—H) infinite point on Euler line |
310 b2/(SBC+SAB–2SCA)
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(SBC+SAB–2SCA)2/b2SB
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b2SB/(SBC+SAB–2SCA)
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(SBC+SAB–2SCA)2/SB
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(SBC+SAB–2SCA)2SB
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? b2(c2–a2)SBB
∞•~H2 |
107 1/(c2–a2)SBB
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2972 b2(c2–a2)2SBBB
intersection with O inconic |
648 1/(c2–a2)SB
? |
? b4(c2–a2)2SBBB
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? b4(c2–a2)2SBBBBB
- |
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Fissile points
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not so sure what to make of these
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? b(c-a)sb(4SBB-c2a2)
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? 1/bsb(4SBB-c2a2)
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655 (c-a)sb2(4SBB-c2a2)2
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The MacBeath circumconic, the dual of the Macbeath inconic
The dual of a point on a circum or inconic is tangent to an in- or circumconic. These conics are called dual. The dual of the MacBeath inconic is the O circumconic. This picture shows points on both and lines that are tangent to the circumconic. The table of points above gives information about them
Figure: The MacBeath inconic is red, the Jerabek circumhyperbola in dark blue and the O circumconic in a lighter blue. The circumcircle, Steiner ellipse, as well as the H and D inconics are also shown. There are lines connecting the most important points. Many of these lines are the natural lines associated with the Jerabek hyperbola.
Strong points
The strong points lies on lines connecting other well known points on the Steiner ellipse, the circumcircle and the Jerabek hyperbola as shown in the following two pictures.
Figure: Strong points on the MacBeath inconic (red)
