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K026 and K044 are two central circum-cubics with center X(5) passing through X(3), X(4) and the centers of the Johnson circles i.e. the reflections of A, B, C in X(5).

They generate a pencil of central cubics passing through these same 9 points. One cubic decomposes into the Euler line and the circum-conic with center X(5). This is one of the three nKs of the pencil which also contains three psKs. One of them is in fact a pK namely K044.

Every cubic (K) of the pencil meets the line at infinity and the circumcircle again at the same points as two isogonal pKs with pivots P1, P2 respectively.

P1 lies on the line (L1) passing through X(i) for i = 3, 54, 97, 195, 1154, 1157, 1493, 1993, 2979, 5012, 5889, 5890, 6101, 6102, 6150, 7592, 7691, 8883, 10203, 10574, 10610, 10627, 11126, 11127, 11402, 11412, 11422, 11423, 12060, 12160, 12161, 12307, 12316, 12363, 12606, 13409, 13630, 15087, 15135, 15137, 15345, 15801, 16030, 16035, 16266, 16762, 18016, 19167, 19168, 19170, 19194, 19209, 19210, 19211, 20791, 22815, 23061, 23606, 25042, 25044, 27246, 31388, 31807, 31810, 32046, 32078, 32136, 32333, 32338, 32339, 32341, 32608, 34148, 34292, 34396, 34424, 34425, 34553, 34555, 34833, 35195, 35196, 35449.

P2 lies on the line (L2) passing through X(i) for i = 5, 49, 54, 110, 265, 567, 1141, 3615, 4993, 6288, 7604, 8254, 8836, 8838, 8901, 9705, 9706, 10211, 10272, 11597, 11801, 11804, 12022, 12026, 12228, 13434, 14389, 14516, 14643, 14644, 14674, 14769, 15089, 15367, 15425, 15426, 15806, 18350, 18464, 18883, 19176, 19193, 20584, 20585, 23236, 25339, 27196, 27423, 31656, 31675, 32410, 32423, 32638, 34308, 34596, 34597, 34770, 36966.

This pencil contains some other remarkable cubics as shown in the table below.

(K)

type

P1

P2

centers X(i) on (K) apart X(3), X(4), X(5) for i =

K026

psK60

X(3)

X(5)

5403, 5404, 8798, 14363

K044

pK

X(5889)

X(14516)

52, 68, 155, 5562, 8800, 8905, 8906, 34428

K465

circular

X(1154)

X(32423)

1154, 1157, 1263, 14072, 14979, 14980, 19552, 24772, 33565

K526

spK

X(54)

X(6288)

54, 6288, 25043

K562

psK

X(34148)

{4,52}

 

nK(X5 x X2574, X2593, X3)

nK

?

?

2574

nK(X5 x X2575, X2592, X3)

nK

?

?

2575

K1181

-

X(5012)

{4,69}

6, 311, 1352

 

-

X(20791)

{2,154}

51, 1073

 

-

X(1993)

{4,193}

69, 216, 1351

 

-

X(2979)

X(12022)

53, 66

 

-

?

?

32, 68, 155

 

-

?

?

316, 671, 2080

note : when a point P1 or P2 is unlisted in ETC, a pair {i,j} means that the point also lies on the line passing through X(i), X(j).

 

A related parabola
table76parabole

The line (L) passing through the two points P1, P2 mentioned above is tangent to a parabola (P) for every cubic of the pencil.

The focus F of (P) is the intersection of the lines {5,930}, {54,1511}, SEARCH = 4.00309039433494.

The infinite point of (P) is that of the lines {5,195}, {54,140}, SEARCH = 9.96240035774153.

(P) is tangent to the Euler line, (L1), (L2) at X(140), X(7691), X(6288) respectively.

Hence (P) is the inconic of triangle X(3)X(5)X(54) whose perspector is Q on the lines {2,568}, {3,161}, {5,7691}, {54,140}, SEARCH = 6.95274277471041.

Q is the reflection of X(54) about the centroid of the triangle and then, it lies on its Steiner ellipse.

Obviously, F lies on the circumcircle of the triangle whose center lies on {54,526},{140,523}, SEARCH = -0.599644778355805.

All the centers mentioned above are unlisted in the current edition of ETC (2021-01-28).