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These two triangles are very frequently mentioned specially in pages devoted to equilateral cubics. Their vertices lie on many interesting curves as seen below.

First we recall some of their properties as in TCCT p.166.

CircumTangential Triangle

This triangle T1T2T3 is formed by the points on the circumcircle such that the line passing through one point and its isogonal conjugate is tangent to the circumcircle.

It is equilateral and homothetic to the Morley triangle.

The homothety h(X20, 3/2) maps its vertices to the cusps of the Steiner deltoid.

T1, T2, T3 lie on the Kjp cubic and other curves as seen below.

CircumNormal Triangle

This triangle N1N2N3 is formed by the points on the circumcircle such that the line passing through one point and its isogonal conjugate is normal to the circumcircle i.e. passes through O.

It is equilateral and homothetic to the Morley triangle.

The homothety h(X631, -3/2) maps its vertices to the cusps of the Steiner deltoid.

N1, N2, N3 lie on the McCay cubic and other curves as seen below.

***

These CircumTangential and CircumNormal triangles are symmetric in O.

See also the related FG paper "A Morley configuration".

 

Related Conics

Any conic passing through O and the vertices of one of these two triangles is obviously a rectangular hyperbola. This gives two pencils of rectangular hyperbolas we call CircumTangential and CircumNormal rectangular hyperbolas. Each conic of each pencil is entirely characterized by a fifth point P = u:v:w on the curve. Their equations are rather complicated and can be downloaded here :

The following tables give a selection (some of them highlighted) of such hyperbolas with given P on the circumcircle.

CircumTangential rectangular hyperbolas

CircumNormal rectangular hyperbolas

P

centers on the curve

X(74)

X(3), X(74), X(2574), X(2575)

X(98)

X(2), X(3), X(98), X(3413), X(3414)

X(102)

X(3), X(40), X(102)

X(103)

X(3), X(103), X(165)

X(104)

X(3), X(21), X(104)

X(105)

X(3), X(55), X(105)

X(106)

X(1), X(3), X(106)

X(111)

X(3), X(111), X(574), X(1995)

X(477)

X(3), X(30), X(477), X(523)

X(699)

X(3), X(32), X(699)

X(713)

X(3), X(713), X(1333)

X(727)

X(3), X(58), X(727)

X(729)

X(3), X(6), X(729)

X(733)

X(3), X(39), X(733)

X(741)

X(3), X(171), X(741)

X(759)

X(3), X(35), X(759)

X(840)

X(3), X(840), X(1155)

X(842)

X(3), X(23), X(842)

X(915)

X(3), X(28), X(915)

X(917)

X(3), X(27), X(917)

X(933)

X(3), X(933), X(1624)

X(934)

X(3), X(934), X(2283)

X(953)

X(3), X(513), X(517), X(859), X(953)

X(972)

X(3), X(972), X(1817)

X(1141)

X(3), X(5), X(1141)

X(1289)

X(3), X(1289), X(2409)

X(1294)

X(3), X(20), X(1294)

X(1297)

X(3), X(22), X(1297)

X(1298)

X(3), X(418), X(1298), X(2979)

X(1299)

X(3), X(24), X(1299)

X(1300)

X(3), X(4), X(1300)

X(1477)

X(3), X(57), X(1477)

X(2371)

X(3), X(9), X(2371)

X(2687)

X(3), X(1325), X(2687)

X(2693)

X(3), X(2071), X(2693)

X(2697)

X(3), X(858), X(2697)

X(2698)

X(3), X(237), X(511), X(512), X(2698)

X(2716)

X(3), X(2077), X(2716)

X(2718)

X(3), X(36), X(2718)

X(2724)

X(3), X(514), X(516), X(2724)

X(2734)

X(3), X(515), X(522), X(2734)

X(2757)

X(3), X(8), X(2757)

P

centers on the curve

X(99)

X(3), X(99), X(376), X(3413), X(3414)

X(100)

X(3), X(100)

X(107)

X(3), X(4), X(107), X(1075)

X(108)

X(3), X(56), X(108)

X(109)

X(1), X(3), X(109), X(1745)

X(110)

X(3), X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575)

X(112)

X(3), X(32), X(112), X(378)

X(476)

X(3), X(30), X(476), X(523)

X(691)

X(3), X(691), X(2080)

X(805)

X(3), X(511), X(512), X(805)

X(901)

X(3), X(513), X(517), X(901)

X(925)

X(3), X(20), X(925)

X(927)

X(3), X(514), X(516), X(927)

X(930)

X(3), X(550), X(930)

X(934)

X(3), X(934), X(999)

X(1291)

X(3), X(1157), X(1291)

X(1293)

X(3), X(40), X(1293)

X(1301)

X(3), X(24), X(1301)

X(1302)

X(2), X(3), X(1302)

X(1304)

X(3), X(186), X(1304)

X(1309)

X(3), X(515), X(522), X(1309)

X(2222)

X(3), X(36), X(2222)

X(2715)

X(3), X(1691), X(2715)

X(2720)

X(3), X(1319), X(2720)

X(2731)

X(3), X(944), X(2731)

X(2743)

X(3), X(2077), X(2743)

X(2867)

X(3), X(525), X(1503), X(2867)

Remark 1 : the most interesting hyperbola is probably the CircumNormal rectangular hyperbola passing through X(110) since it contains eight ETC centers.

Its center is X(1511) and its asymptotes are parallel to those of the Jerabek hyperbola.

We meet this hyperbola in table 16 : it is the locus of points whose polar conic in the Neuberg-Lemoine pencil is a circle.

***

Remark 2 : the yellow, light blue, orange cells correspond to hyperbolas homothetic to the Jerabek, Kiepert, Feuerbach hyperbolas respectively.

Note that the inversive image in the circumcircle of any of the rectangular hyperbolas gives a strophoid passing through the vertices of one of the two triangles. See K725 for example which is the inverse of the hyperbola of the remark above.

The CircumTangential rectangular hyperbolas are the images of lines passing through G under the involution f defined in the page K024.

 

Cubics

The following tables give a selection of remarkable cubics through the vertices of these triangles.

CircumTangential cubics

CircumNormal cubics

cubic

other centers on the curve

K024

none

K078

X(1), X(2), X(3), X(165), X(5373)

K085

X(1)

K098

none

K105

X(3), X(4)

K403

none

K409

none

K686

X(2), X(6)

K723

X(3)

K726

X(6)

K727

X(2), X(3)

K728

X(2), X(3), X(23), X(111), X(187), X(2930), X(3098)

K734

X(2), X(3), X(4230)

K735

X(3), X(182), X(3098), X(8666), X(8715)

K896

X(6), X(111), X(368), X(511), X(3098), X(5640), X(6194)

cubic

remark / other centers on the curve

K003

McCay cubic

K009

Lemoine cubic

K227

only one vertex on each curve

K361

X(3), X(4), X(54), X(1342), X(1343)

K373

X(3), X(54), X(96), X(1147)

K404

nK0(X924, X323)

K405

McCay-Lemoine cubic. D(1/2), see Table 33

K519

an Orion cubic. See Table 11

K664

isogonal transform of K665

K725

Neuberg strophoid

K735

X(3), X(182), X(3098), X(8666), X(8715)

K736

X(3), X(40), X(376), X(7709)

See also Table 54 for a generalization.

pK cubics

The loci of poles and pivots of CircumNormal pKs are K378 and K361 respectively. The locus of isopivots is K405.

The loci of poles and pivots of CircumTangential pKs are K402 and K403 respectively.

nK0 cubics

The CircumTangential nK0s must have their pole either

– on the circumconic with perspector X(32) in which case the cubic decomposes into the circumcircle and a line isoconjugate of the circumcircle. The root must lie on the circumcircle.

– on the Brocard axis. The root lies on the line GK and the line passing through X(110) and the pole. See Kjp = K024 for example, also nK0(X511, X323).

The CircumNormal nK0s must have their pole either

– on the circumconic with perspector X(32) in which case the cubic decomposes into the circumcircle and a line isoconjugate of the circumcircle. The root must lie on the circumcircle.

– on the line X(50)X(647). The root lies on the line X(323)X(401). These two lines meet at X(2623), the crossconjugate of X(115) and X(6). K404 = nK0(X924, X323) is an example of such cubic.

psK cubics

For any pseudo-pole Ω, one can always find one CircumNormal psK and one CircumTangential psK with respective pseudo-pivots :

a(X311 x Ω) x Ω ÷ X6 and a(X850 x Ω) x Ω ÷ X6,

where x and ÷ are the barycentric product and quotient, aX is the anticomplement of X.

Similarly, for any pseudo-pivot P, one can always find one CircumNormal psK and one CircumTangential psK with respective pseudo-poles :

a(X5 x P) x P x X6 and a(X523 x P) x P x X6.

These psKs become pKs when Ω and P lie on the loci mentioned above.

 

Higher degree curves

The following tables give a selection of remarkable curves through the vertices of the triangles.

CircumTangential curves

CircumNormal curves

curve

name

Q031

a nonic

Q046

McCay butterfly

CL004

isogonal nK60 cubics

 

 

 

 

curve

name

Q007

an octic

Q009

a bicircular septic

Q010

a central circular quintic

Q018

a bicircular octic

Q020

a tricircular octic

Q023

an inversible quartic

Q046

McCay butterfly

Q047

McCay quartic