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This page has been written using several insightful ideas by JeanPierre 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 

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. 

gamma_1 is the "sum" of the three conics gamma_1x seen above. Its equation is : Sqrt3 (C)  2S (G) = 0 where :
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. 



Cubics passing through the equilateral cevian points 

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



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 circumcubics 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 : 


