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K765

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X(2), X(3), X(3524), X(5024), X(5373), X(5646)

Q1, Q2, Q3 : vertices of the Thomson triangle T

A', B', C' : excenters of T, the incenter being X(5373). The tangents at these four points concur at X(3524)

M1, M2, M3 : midpoints of T with tangents passing through X(3)

H1, H2, H3 : midpoints of the altitudes of T, on the sidelines of M1M2M3

infinite points of the cubic pK(X6, P) where P is the reflection of X(3524) in X(3) i.e. X(376) of T.

K765 is the Thomson cubic K002 of the Thomson triangle T = Q1Q2Q3. Hence, it is a member of the Euler pencil of cubics in T, see K764.

It follows that K765 passes through the counterparts of all the points of K002 in T.

K765 meets the circumcircle at Q1, Q2, Q3 and three other points lying on nK0(X6, S) where S is X(14930), a point on the line X(2)X(6).

K765 meets the line at infinity at the same points as pK(X6, P) as above. The six remaining (finite) common points lie on the rectangular hyperbola (H) passing through X(3), X(6), X(376), X(1992), X(2574), X(2575) hence homothetic to the Jerabek hyperbola.

K765 and K002 meet at Q1, Q2, Q3 (with the same tangents passing through X(5646), the Lemoine point of T), X(2) also with the same tangent namely the line X(2)X(6) and at last X(3).