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K1207

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X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(30), X(41975), X(41976), X(42725), X(42726), X(42727), X(42728), X(42729), X(42730)

Geometric properties :

K1207 is the KHO-cubic obtained when P = X(4). See K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (y - z) - z (2y - z) (y + z) = 0.

X(6) is a point of inflexion with inflexional tangent passing through X(5) and harmonic polar the Euler line.

Hence, a line passing through X(6) meets the Euler line at E and K1207 at P1, P2 which are harmonic conjugates with respect to X(6) and E.

Furthermore, the midpoint of P1, P2 lies on a hyperbola (H) passing through X(2), X(4), X(6), X(115), X(187), X(1506), X(1560), X(2039), X(2040), etc.

Note that the infinite points of (H) lie on K1207.

K1207 is bitangent to the Kiepert hyperbola at G, H and the remaining common points are X(13), X(14).

K1207 meets the Evans conic at X(13), X(14), X(15), X(16), X(41975), X(41976).

***

More generally, every pivotal KHO-cubic K(P) with pivot P on the van Aubel line (through H and K) passes through X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(30). Conversely, every cubic passing through these eight points is a pivotal KHO-cubic of this type.

These cubics are therefore in a same pencil generated by K1207 and the union of the Euler, Brocard, Fermat lines.

For instance, with P = X(397), X(398), X(1503), we find the cubics K1221a, K1221b, K1223. See also K1230.

If P has KHO-coordinates (1,T,0), the KHO-equation of K(P) is : T [x^2 (y - z) - z (2y - z) (y + z)] - y z (y - 2z) = 0.

When T = 0, ±1, ± 1/2, hence P = X(6), X(5318), X(5331), X(5334), X(5335), K(P) decomposes.

Other KHO-points on K(P) :

• Q1 = (3,2T, - 2T), on the line X(6), X(30).

• Q2 = (3,T,2T), on the line X(2), X(6).

• Q3 = (2T,1,1), on the line X(5), X(6). Q3 is the isopivot hence the common tangential of X(2), X(6), X(30) and P.

• Q4 is the common tangential of X(4), Q1, Q2 and Q3. Q4 = ( - 2 T (4 T^2 - 5),4 T^2 - 1,4 T^2 - 5).

• Q5 = (8 T,( - 1 + 2 T) (1 + 2 T),4)

• Q6 = (3 (3 + 4 T^2), - 2 T ( - 3 + 2 T) (3 + 2 T),2 T (3 + 4 T^2))

• Q7 = (3 ( - 3 + 4 T^2), - T ( - 3 + 2 T) (3 + 2 T), - 2 T ( - 3 + 4 T^2))

• Q8 = (12 T ( - 5 + 4 T^2),( - 3 + 2 T) (3 + 2 T) ( - 3 + 4 T^2), - ( - 3 + 4 T^2) (3 + 4 T^2))

• Q9 = (24 T ( - 5 + 4 T^2),( - 3 + 2 T) (3 + 2 T) (3 + 4 T^2),2 ( - 3 + 4 T^2) (3 + 4 T^2))

• Q10 = (( - 3 + 4 T^2) (3 + 4 T^2),4 T ( - 1 + 2 T) (1 + 2 T),4 T ( - 5 + 4 T^2))

• Q11 = (( - 3 + 4 T^2) (3 + 4 T^2), - T ( - 1 + 2 T) (1 + 2 T) ( - 5 + 4 T^2), - 4 T (-5 + 4 T^2))

Triads of collinear points on K(P) :

X2 – X4 – X30 || X2 – X6 – Q2 || X2 – X13 – X16 || X2 – X14 – X15 || X2 – P – Q1 || X2 – Q4 – Q9 || X2 – Q5 – Q8

X2 – Q6 – Q11 || X2 – Q7 – Q10 || X4 – X6 – P || X4 – Q1 – Q7 || X4 – Q2 – Q6 || X4 – Q3 – Q5 || X6 – X13 – X14

X6 – X15 – X16 || X6 – X30 – Q1 || X6 – Q4 – Q10 || X6 – Q5 – Q11 || X6 – Q6 – Q8 || X6 – Q7 – Q9 || X13 – X15 – X30

X14 – X16 – X30 || X30 – P – Q2 || X30 – Q4 – Q8 || X30 – Q5 – Q9 || X30 – Q6 – Q10 || X30 – Q7 – Q11 || P – Q4 – Q11

P – Q5 – Q10 || P – Q6 – Q9 || P – Q7 – Q8 || Q1 – Q2 – Q5 || Q1 – Q3 – Q6 || Q2 – Q3 – Q7

KHO-equations of the loci :

• Q4 : x^2 (y - z) - (5 y - z) z^2 = 0, K1224, a crunodal cubic through {X4,X6,X13,X14,X15,X16,X549}

• Q5 : x^2 - z (4 y + z) = 0, a hyperbola through {X4,X13,X14,X15,X16,X376,X3068,X3069}

• Q6 : x^2 (y - 3 z) + 3 z^2 (y + z) = 0, K1225, an acnodal cubic through {X4,X6,X13,X14,X15,X16,X30,X485,X486}

• Q7 : x^2 (2 y - 3 z) - 3 (2 y - z) z^2 = 0, K1226, a crunodal cubic through {X2,X4,X6,X13,X14,X15,X16}

• Q8 : 3 x^2 (y - z)^2 + (y - 3 z) (2 y - z)^2 (y + z) = 0, a quartic through {X2,X6,X13,X14,X15,X16,X30,X546}

• Q9 : 3 x^2 (y - z)^2 - (2 y - 3 z) (2 y - z) (y + z)^2 = 0, a quartic through {X2,X6,X13,X14,X15,X16,X30,X3091}

• Q10 : x^2 (y - z) (5 y - z) - (2 y - z)^2 (y + z)^2 = 0, a quartic through {X2,X6,X13,X14,X15,X16,X30}

• Q11 : (2 y - z)^2 z (y + z)^2 - x^2 (y - z)^2 (4 y + z) = 0, a quintic through {X2,X4,X6,X13,X14,X15,X16,X30}