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K1236

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X(6), X(13), X(14), X(15), X(16), X(41100), X(41101), X(44015), X(44016), X(44017), X(44018), X(44019), X(44020)

other points below

Geometric properties :

K1236 is a KHO-cubic, see K1191 for explanations and CL075.

Its KHO-equation is equivalently

(1) : (y - 2 z)^3 + (y + 2 z) (x^2 - 6 y z + 3 z^2) = 0, where x^2 - 6 y z + 3 z^2 = 0 is the Kiepert hyperbola,

(2) : 2 y (- x + y + z) (x + y + z) + (y - 2 z) (x^2 - 3 y^2 + 2 y z - z^2) = 0, where x^2 - 3 y^2 + 2 y z - z^2 = 0 is the Evans conic.

K1236 has a triple contact with the Kiepert hyperbola at X(13) and X(14). The tangents at these points are parallel to the Euler line and their tangentials are X(15), X(16) respectively.

K1236 is bitangent at X(15), X(16) to the Evans conic and the tangents at these points pass through X(5).

X(6) is a point of inflexion with tangent passing through X(382).

The tangentials of X(15), X(16) are Q1 = (-5,6,1), Q2 = (5,6,1) respectively. Q1 lies on the lines {X5,X16},{X15,X3146},{X17,X1657} and Q2 on {X5,X15},{X16,X3146},{X18,X1657}.

Q3 = (9,2,-5) and Q4 = (9,-2,5) are two simple points on K1236 and on the line through X(6), X(3534). Q3 also lies on the lines {X2,X13},{X18,X381},{X30,X62},{X61,X376} and Q4 on {X2,X14},{X17,X381},{X30,X61},{X62,X376}.

These four points are X(44016), X(44015), X(41100), X(41101) respectively.

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Generalization

Every KHO-cubic with equation (y - 2 z)^3 + (v y + w z) (x^2 - 6 y z + 3 z^2) = 0 has a triple contact with the Kiepert hyperbola at X(13) and X(14).

One remarkable example is obtained with (v,w) = (5,2) since the cubic contains X(6), X(16241), X(16242) and six simple points on the Evans conic, namely X(13), X(14), X(15), X(16), X(590), X(615).

With (v,w) = (-5/3,4/3), the cubic contains X(5), X(6), X(13), X(14), X(30), X(371), X(372), X(3830), X(32787), X(32788).