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This page has been written using several insightful ideas by Jean-Pierre Ehrmann

A point P is an equilateral cevian point if the cevian triangle of P is equilateral (TCCT, p.267). The corresponding center of the triangle is denoted by Q. There are 6 such points P which lie on the Neuberg cubic: 4 are always real, one of them being X(370), the only one inside ABC. Naturally, there are also 6 corresponding points Q which lie on the Napoleon cubic and on the perpendicular bisectors L1 and L2 of X(13)X(15) and X(14)X(16) respectively. Those on L2 (blue corresponding points P ) are always real but those on L1 (red corresponding points P) are not necessarily : the always real point on L1 is associated to X(370). Each line through P and the corresponding Q passes through O. See also Q033.

These 6 points P (and their isogonal conjugates P*) lie on a large number of curves which are shown below.

See also table 14 and table 40 when "cevian" is replaced with "anticevian" and "circumanticevian".

 

Conics and circles passing through the equilateral cevian points

Two groups of three conics

gamma_1a and gamma_2a are the conics with equations :

Sqrt3[S_B y(x+z)+S_C z(x+y)] - 2S x(y+z+2x) = 0

Sqrt3[S_B y(x+z)+S_C z(x+y)] + 2S x(y+z+2x) = 0

which pass through B, C, A' (reflection of A in BC), Ga (S is the area of ABC).

The lines passing through A' and the third vertices of the two equilateral triangles erected externally on AB, AC meet the lines AB, AC at two points on gamma_1a. For gamma_2a, replace externally by internally.

The remaining four conics are constructed similarly.

The conics gamma_1x contain the red points, and the conics gamma_2x the blue points.

A group of two conics

gamma_1 is the "sum" of the three conics gamma_1x seen above. Its equation is :

Sqrt3 (C) - 2S (G) = 0 where :

  • (C) = a^2yz + cyclic
  • (G) = x^2 + yz + cyclic

showing that gamma_1 belongs to the pencil of conics generated by the circumcircle and an imaginary ellipse with center G, homothetic of the Steiner ellipse. gamma_1 contains the red points and X616.

Similarly, gamma_2 has equation :

Sqrt3 (C) + 2S (G) = 0, and contains the blue points and X617.

In other words, the conics gamma_1x generate a net N_1 of conics which contains gamma_1.

The figure shows two of the six equilateral cevian triangles.

A group of two circles

The net N_1 seen above also contains a circle which is Gamma_1. Gamma_2 is defined similarly.

These two circles pass through D (reflection of H in X110) and E (reflection of D in the line X20-X110). Their centers lie on this line X20-X110.

E is X(5667) = X(30)-Ceva conjugate of X(4).

Gamma_1 and gamma_1 meet at the red points and at Q1. Gamma_2 and gamma_2 meet at the blue points and at Q2. Q1 and Q2 lie on the line X74-X99.

The equations are rather complicated. Click on the link to download a text file :

A group of two circles

Similarly, it is not difficult to find analogous groups of six conics, two conics and two circles passing through the isogonal conjugates P* of the 6 points P. These points P* obviously also lie on the Neuberg cubic.

We only show the group of circles Gamma_1*, Gamma_2* and the group of conics gamma_1*, gamma_2* (beware these curves are NOT the isogonal conjugates of the ones met before).

Gamma_1*, Gamma_2* are both centered on the line through O, X74, X110, X399 (this latter point lying on both circles).

Gamma_1* and gamma_1* intersect at the isogonal conjugates of the red points (orange points) and meet again at a point on the line OX110. Similarly, Gamma_2* and gamma_2* intersect at the isogonal conjugates of the blue points (light-blue points) and also meet again at a point on the line OX110.

Click on the link to download a text file of all the ten conics here mentioned.

 

Cubics passing through the equilateral cevian points

The table below sums up different cubics through these red and blue points P :

cubic

name

points

other points

K001

Neuberg cubic

P, P, P*, P*

see the Neuberg page

K143-A-B-C

X(370)-cubics

P, P

Ga, Gb, Gc

K144

X(370)-antimedial cubic

P, P

X(2), X(3), X(30), X(5667), Ga, Gb, Gc

K264a

pK(X2, X298)

P

X(2), X(13), X(298), X(616)

K264b

pK(X2, X299)

P

X(2), X(14), X(299), X(617)

K265

X(370)-medial cubic

P, P

midpoints

K266

X(370)-equilateral cubic

P, P

 

K341a

pK(X15, X2)

P

X(2), X(3), X(6), X(13), X(15), X(18), X(298), X(470), X(1082), X(2992), X(5240)

K341b

pK(X16, X2)

P

X(2), X(3), X(6), X(14), X(16), X(17), X(299), X(471), X(559), X(2993), X(5239)

K419a

pK(X13, ?)

P

X(2), X(13), X(14), X(30), X(298), X(395), X(472), X(1081)

K419b

pK(X14, ?)

P

X(2), X(13), X(14), X(30), X(299), X(396), X(473), X(554), X(7043)

Remark 1 : there are two cubics which are both decomposed into three lines, each passing through either the red or blue points P. They are shown in the figures above in dashed lines. The equations are complicated.

Remark 2 : K264a, K341a, K419a belong to a same pencil of circum-cubics through the blue points. This pencil contains no other pK and three very complicated and not always real nK0s. The same remark applies to K264b, K341b, K419b and the red points.

 

Higher degree curves passing through the equilateral cevian points

The table below sums up different higher degree curves through these red and blue points P :

curve

name

points

other points

Q033

X(370)-quartic

P, P

X(2), X(3), X(30)

Q034

X(370)-Fermat quintic

P, P

X(2), X(3), X(13), X(14)

Q035-A-B-C

X(370)-central quartics

P, P

Ga, Gb, Gc

Q036-A-B-C

X(370)-central quintics

P, P

Ga, Gb, Gc

Q074

X(370)-Soddy quintic

P, P

X(2), X(4), Soddy centers, X(616), X(617), etc

Q099

X(370)-quartic

P, P

X(3), X(6), X(1249)

Q100

Dergiades septic

P, P

X(2), X(7), X(369), etc

Q105

Montesdeoca septic

P, P

X(1), X(2), X(3), X(1138), etc

Q111

a quintic

P, P

X(3), X(4), X(1263), X(2992), X(2993)