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X(74), X(1511)

other points described below

K374 is the isogonal transform of K037, the Tixier equilateral cubic.

Hence, it meets the circumcircle at A, B, C and three other points X(74), Z1, Z2 which are the vertices of an equilateral triangle.

Z1 and Z2 are isoconjugates on the curve and their barycentric product is X(50). Their midpoint is X(1511). The tangents at these two points and at the pivot X(1511) pass through X(110).

In fact, the polar conic of X(110) is the union of the line X(526)X(X1511) (which contains Z1 and Z2) and the Brocard axis.

Note that the polar conics of the isodynamic points X(15), X(16) are each also decomposed into two lines and these lines are perpendicular. It follows that each point on the Brocard axis has a polar conic which is a rectangular hyperbola.

Notes on X(1511) = Fermat crosssum

– from Clark's ETC

  • X(1511) lies on these lines: 2,265   3,74   24,1112   30,113   36,1464   125,128   141,542   146,376   184,974   186,323   214,960   249,842   389,1493
  • X(1511) = midpoint of X(I) and X(J) for these (I,J): (3,110), (74,399)
  • X(1511) = reflection of X(I) in X(J) for these (I,J): (125,140), (1539,113)
  • X(1511) = complementary conjugate of X(2072)
  • X(1511) = X(I)-Ceva conjugate of X(J) for these (I,J): (3,1154), (110,526)
  • X(1511) = crosspoint of X(I) and X(J) for these (I,J): (2,340), (15,16)
  • X(1511) = crosssum of X(13) and X(14)

– other notes

  • X(1511) is the pole of the Brocard axis in the circum-hyperbola passing through X(2), X(15), X(16), X(186), X(249), X(323), X(842), X(1138), X(2411).
  • X(1511) is the center of the rectangular hyperbola passing through X(3), X(54), X(110), X(182), X(1147), X(1385), the vertices of the circumnormal triangle. Its asymptotes are parallel to those of the Jerabek hyperbola. See Table 16, the Neuberg-Lemoine pencil.