   too complicated to be written here. Click on the link to download a text file.  X(4) H* see below infinite points of the altitudes A', B', C' : vertices of the ITB triangle Geometric properties :   The ITB triangle A'B'C' is defined at K1098. The isogonal conjugation 𝞱 with respect to the ITB triangle swaps A, B, C and the infinite points of the altitudes of ABC. It swaps X(3) and X(194), X(6) and X(7709). An isogonal pK with respect to the ITB triangle is also a circum-cubic of ABC if and only if its pivot is X(4). K1100 is then the corresponding cubic, the locus of M such that X(4), M, 𝞱(M) are collinear but beware it is not a pK with respect to ABC. K1100 is a K+ with asymptotes concuring at X on the Euler line, SEARCH = -10.4930906211322. X = a^2 (a^8 b^2-3 a^6 b^4+3 a^4 b^6-a^2 b^8+a^8 c^2-a^6 b^2 c^2+4 a^4 b^4 c^2-a^2 b^6 c^2-3 b^8 c^2-3 a^6 c^4+4 a^4 b^2 c^4+4 a^2 b^4 c^4+3 b^6 c^4+3 a^4 c^6-a^2 b^2 c^6+3 b^4 c^6-a^2 c^8-3 b^2 c^8) : : = X(32444). The isopivot and tangential of X(4) is H* = 𝞱(X4) on the lines {4,1625}, {4,6}, {32,1671}, etc, SEARCH = -0.888370325987551 with barycentrics a^2 (a^6 b^2-2 a^4 b^4+a^2 b^6+a^6 c^2+a^4 b^2 c^2-a^2 b^4 c^2-b^6 c^2-2 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4+a^2 c^6-b^2 c^6) : : = X(32445). Note that the tangent at X(4) passes through X(6) which is the Lemoine point of ABC and A'B'C'. The polar conic of H* passes through X(4), X(112), X(7709) and obviously A', B', C'. The polar conic (H) of X(4) is a rectangular hyperbola with center X(98) and passes through X(4), X(40), X(376), X(1670), X(1671), X(3413), X(3414) hence it is homothetic to the Kiepert hyperbola. See also K1098. Note that the polar conic in K1100 of any point on the Euler line is a rectangular hyperbola. K1100 and K004 meet at A, B, C, the infinite points of the altitudes of ABC and X(4) counting for three since the tangent and polar conic at X(4) in both cubics are the same. Indeed, K1100 is a member of the pencil of circum-cubics generated by the Darboux cubic K004 and equivalently the union of the altitudes or the union of the line at infinity with the Kiepert hyperbola. 