The Lemoine inellipse
and G K Mineur circumconic
Most conics are defined from their perspector. But an inconic can easily be defined from its two foci, which are isogonally conjugate. The foci of the Lemoine inellipse are the centroid G and symmedian point K. Their midpoint is the center of the conic and from this the perspector can be recovered.
The Lemone conic wins a prize. It has no ETC point on it! It thus ties the Simmons conic (also none) and barely beats out the Mandart inconic (which has 1).See here for a list of ETC points on inconics.
This type of conic is studied differently. Not only does it have no known points on it but it is associated with a circumconic in a unique way.Its foci G and K are isogonal conjugates so that the isogonal conjugate of the line joining them is a Mineur conic, which is guaranteed to have many interesting points as well as to be well connected to the focal inconic.
Figure: The Lemoine conic is blue and its Mineur hyperbola is also blue. The Steiner inellipse, on which the center of the hyperbola lies, is red.The bright red points are the 4 intersections of the Lemoine inconic with the Steiner inconic.
Figure: The intersection of two inconics produces a central point o and three (a, b, c) that are together central and in perspective .
The Lemoine inellipse
This is from the Peter Moses' compilation of points on conics
Lemoine inconic, center 597 4a2+b2+c2, perspector 598 1/(-a2+2b2+2c2)(axis GK line, a strong line)
Peter Moses update creating some points on this conic.
Lemoine InEllipse, center 597, perspector 598. Major axis GK line, Minor axis {597,1499}, Foci G & K.
Semi Major axis = √[(2b2 + 2c2 - a2)(2 c2 + 2 a2 - b2)(2 a2 + 2 b2 - c2)] / (3 SW)
Semi Minor axis = S / √[3/2 SW]
With all this info about, it is quite suprising that the Lemoine InEllipse is quite so ETC desolate
The circle circumscribing the Lemoine InEllipse passes through X(115) & X(125), the centers of the KH and JH.
The line X(115) X(125), the dual of X(99), is the 4th tangent to the Lemoine Inellipse, Orthic Inconic, Simmons inconics and Steiner Inellipse. (inconics with their perspector on the Kiepert circumhyperbola, or equivalently, inconics with center on GK)
This line is tangent to the Lemoine at (b2 - c2)2 (2 b2 + 2 c2 - a2)::,
Orthic at X(125) and Steiner at X(115)
Generally, for an Inconic with perspector P{p,q,r} on the Kiepert circumhyperbola, the line X(115) X(125) is tangent to the inconic at
(b2 - c2) ((a2 - b2) q - (a2 - c2) r) / ((a2 - b2) (a2 - 2 b2 + c2) q + (a2 - c2) (a2 + b2 - 2 c2) r) ::
P{p,q,r} on Circumcircle to Lemoine InEllipse -> a4 / ((2 b2 + 2 c2 - a2) p2) ::
P{p,q,r} at Infinity to Lemoine InEllipse -> p2 / (2 b2 + 2 c2 - a2) ::
For some parameter t, ((b - c)2 (a + t)2) / (2 b2 + 2 c2 - a2) :: is on the Lemoine InEllipse.
Some simple points on Lemoine ...
X(74) on the Circumcircle ->(S2 - 3 SBC)2 / (2 b2 + 2 c2 - a2) ::{{458,598}/\{1383,1989}}
X(100) on the Circumcircle ->a2(b - c)2 / (2 b2 + 2 c2 - a2) ::
X(101) on the Circumcircle ->(b - c)2 / (2 b2 + 2 c2 - a2) ::
X(106) on the Circumcircle ->(2 a - b - c)2/ (2 b2 + 2 c2 - a2) ::
X(110) on the Circumcircle ->(b2 - c2)2 / (2 b2 + 2 c2 - a2) :: {{6,598,671}/\{115,2793(tangent line)}/\{338,1648}/\{1383,1989}
X(111) on the Circumcircle ->(b2 + c2 - 2 a2)2 / (2 b2 + 2 c2 - a2) :: {{2,67}/\{6,598,671}..}
X(112) on the Circumcircle ->(b2 - c2)2 SA2 / (2 b2 + 2 c2 - a2) :: {{2,67}/\{458,598}/\{338,1648}..}
a2 / ( (2 b2 + 2 c2 - a2) (b2 - c2)) :: {{32,111},{74,182},{98,381,598}..} on the Circumcircle ->
(b2 - c2)2(2 b2 + 2 c2 - a2) :: {{2,353},{115,125(tangent line)}/\{338,850}/\{353,2502}/\{1501,1853}}
(2 b2 + 2 c2 - a2) / (b2 - c2):: {{110,892}/\{598,843}..} on the Circumcircle ->
a4 (b2 - c2)2/(2 b2 + 2 c2 - a2)2 :: (Lemoine/Brocard InEllipse tangent point)
(b2 - c2)2(2 b2 + 2 c2 - a2) (b4 + c4 - a4 - b2 c2)2 :: {338,850} (Lemoine/McBeath inconic tangent point)
(2 b2 + 2 c2 - a2)(4 a4 - a2 b2 + b2 - a2 c2 - 4 b2 c2 + c4)2 :: {2, 353} (Lemoine/Kiepert inparabola tangent point)
(b - c)2 (2 b2 + 2 c2 - a2)(4 a2 + b2 + c2 - 3 a b - 3 c a)2 ::
a2 (b - c)2 (2 b2 + 2 c2 - a2)(2 a2 - b2 - c2 - 3 b c)2 ::a4 (2 b2 + 2 c2 - a2) (3 (b4 + c4 - a4) - b2 c2 (J2 + 3))^2 :: , J = OH/R
Here is a link to my value added version of Peter's compilations.
The G, K Mineur conic
Any inconic, whose foci are isogonal conjugates, has an associated Mineur circumconic, defined by the foci. In our case this is the circumconic through G and K.
This hyperbola is one of a set based on its anallagmatic complement, the Kiepert hyperbola, and the K—R Mineur conic, which embodies the symmetry of both. Hopefully I will write about this someday.
Figure: The Lemoine conic is Green, the Steiner inellipse, blue and the GK conic a dark blue.
Points on Lemoine and GK conics
This is from the Peter Moses' compilation of ETC points on conics The colors I have added. Blue represents strong, red weak, green fissile. Light colors are points that I do not consider very significant and will ignore . See here for ETC.
A,B,C,G,K. center 1084 a4(b2-c2)2, perspector 512 a2(b2-c2) Mineur G K, perspector at infinity
{2 , 6 (K), 25 (pH), 37 mtIo , 42 (pSo),111 g(∞•(G—K), 251 gmK,263, 308 rmK, 393 1/SB2 , 493, 494, 588,589 (these are geometrically defined as centers of homothety), 694 gK2– = a2/a4–, 941 {a/(a^2+2bc+ab+ac)}, 967 a2/(2asa+2sbc+a2), 1169 {a2/(b + c)(b2+ +c2+)}, 1171 {a2/(b + c)(2a+b+c)}, 1218, 1239,1241{1/a2( b4+ + c4+)}, 1383 g(∞•(G—K), 1400 a(b+c)/sa, 1427 a(b+c)/sa2, 1880 {a (b+c) / sa SA}, 1976 pT, 1989 1/(SA2–b2c2), 2054, 2165 {1 / (4S2–b2c2) ,2248, 2350 a2/(asa+2bc), 2395 {(b2-c2)/(b4-+c4-)}, 2433{a2(b2-c2)/(a2SA-2SBC)}, 2963, {1/(4S2-b2c2)}, 2981, 2987{a2/(a4-2SBC-a2SA), 2998 tdR=1/(–b2c2+c2a2+a2b2)
Figure: points on the G—K circumconic. Strong points are blue, weak are red, fissile, green. Also shown is the Lemoine inconic. The Lemoine parabola in red, whose perspector is rS and focus is S, also shown. "from 668" means that this point is projected from the infinite point 668 (see here).
Figure: Lemoine and its Mineur hyperbola with the Steiner ellipse and the circumcircle, showing that the asymptotes of the Mineur conic go through the vertices of the ellipse.
Lemoine Parabola
Any circumconic whose perspector is at infinity, as is that of the GK hyperbola has a dual inparabola. For this conic its perspector is rS, focus S, directrix HR
4th tangents
blah
Figure: inconics tangent to ~S, the dual of S. Strong conics, the Lemoine, the dual circumcircle, and the H inconic, are blue. The four Spieker inconics are red. The Simmons conics are green. Their centers are on GK and the perspectors on the Kiepert hyperbola. [I learned this from Peter Moses]. Wilson Stothers has commented on the surprising number of circumconics thorugh S, which is equivalent to these inconics tangent to the dual of S.
Figure: The Leomoine inellipse is tangent to the MacBeath and Brocard inellipses.