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X(2), X(6), X(25), X(614), X(7735), X(40131), X(45141)

X(46344) → X(46348)

vertices of the Thomson triangle

points of nK0(X6, R2) on (O), see note below

pairs {M1,M2}, {N1,N2} defined in Table 78

points of pK(X6, X26255) at infinity

Geometric properties :

K1252 is introduced in Table 77.

Let R be a point on K1252. The cubic K(R) passing through the anti-points (see Table 77) and the points Q1, Q2, Q3 (apart A, B, C) of nK0(X6, R) on (O) meets the circumcircle again at three points of an isogonal pK with pivot P1 on K002 and the line at infinity at three points also on an isogonal pK with pivot P3. Note that the antipodes of Q1, Q2, Q3 on (O) also lie on an isogonal pK with pivot P2 on K004.

Example : when R = X(6), nK0(X6, R) is K024, K(R) is K727 and the three pKs above are K002, pK(X6, X3534), K003 respectively.

X(6) is a point of inflexion with tangent passing through X(373), X(2434). K1252 is tangent at G to the Euler line.

The tangents to K1252 at the vertices Q1, Q2, Q3 of the Thomson triangle concur at X(11284), intersection of the Euler line and the inflexional tangent above. It follows that the polar conic of X(11284) in K1252 must contain X(2), X(6), Q1, Q2, Q3 and then, it is the Thomson-Jerabek hyperbola. Moreover, K1252 is a psK in the Thomson triangle since X(11284) is not on the cubic.

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Note

R2 = a^6+a^4 b^2-a^2 b^4-b^6+a^4 c^2-10 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6 : : , SEARCH = 4.03166518521035, on the Euler line.

The antipodes of the points (apart A, B, C) of nK0(X6, R2) on (O) lie on pK(X6, S2) where

S2 = 5 a^10-5 a^8 b^2-14 a^6 b^4+22 a^4 b^6-7 a^2 b^8-b^10-5 a^8 c^2+72 a^6 b^2 c^2-46 a^4 b^4 c^2-24 a^2 b^6 c^2+3 b^8 c^2-14 a^6 c^4-46 a^4 b^2 c^4+62 a^2 b^4 c^4-2 b^6 c^4+22 a^4 c^6-24 a^2 b^2 c^6-2 b^4 c^6-7 a^2 c^8+3 b^2 c^8-c^10 : : , SEARCH = 22.6414479780346, on the line {20,64}.

These points are X(46336), X(46349) in ETC.

Other points on K1252

Z1 = a (a^6-2 a^5 b+a^4 b^2-a^2 b^4+2 a b^5-b^6-2 a^5 c-2 a^4 b c+4 a^2 b^3 c+2 a b^4 c-2 b^5 c+a^4 c^2-6 a^2 b^2 c^2-4 a b^3 c^2+b^4 c^2+4 a^2 b c^3-4 a b^2 c^3+4 b^3 c^3-a^2 c^4+2 a b c^4+b^2 c^4+2 a c^5-2 b c^5-c^6) : : , SEARCH = 0.906568395174829, on the lines {2,40131}, {614,45141}.

Z2 = a (a^6-a^4 b^2-a^2 b^4+b^6+6 a^4 b c-4 a^3 b^2 c-4 a b^4 c+2 b^5 c-a^4 c^2-4 a^3 b c^2+2 a^2 b^2 c^2+4 a b^3 c^2-b^4 c^2+4 a b^2 c^3-4 b^3 c^3-a^2 c^4-4 a b c^4-b^2 c^4+2 b c^5+c^6) : : , SEARCH = 0.818664977994983, on the lines {25,40131}, {614,7735}.

Z1, Z2 are X(46345), X(46344) in ETC.