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X(1), X(1498), X(3346)


Geometric properties :

K1190 is a tri-isogonal cubic. See Bi-isogonal and Tri-isogonal Pivotal Cubics. See also the analogous cubics K701 and K719.

K1190 and its Hessian share the same orthic line , namely the line (L) passing through X(i) for i = 3, 64, 154, 1073, 1498, 1619, 2328, 2360 and many others.

K1190 and (L) meet at X(1498) and two other (not always real) points P1, P2 with tangentials Q1, Q2 respectively.

P1, P2 are two isogonal conjugate points on (L) passing through X(3) hence they lie on the McCay cubic K003 and on the rectangular circum-hyperbola (H), the isogonal transform of (L).

Q1, Q2 are two X(1498)-Ceva conjugate points on the line (L') passing through X(2) and X(3346), the isogonal conjugate of X(1498) and also its tangential.

K1190 is an isogonal pK in ABC but also in two other triangles T1, T2.

For i = 1 or 2, the vertices of Ti are the three common points (apart Pi, Qi) of K1190 and the (green) polar conic of Qi in K1190. The in/excenters of Ti are the four common points (apart Pi) of K1190 and the (blue) polar conic of Pi in K1190.

K1190a K1190b