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X(3), X(30), X(74), X(10745)

T1, T2, T3 : vertices of the CircumTangential triangle

R1, R2, R3 : points on its sidelines and on the perpendicular bisector of X(3)X(74)

Va, Vb, Vc described in the page K001

(H) is the rectangular circum-hyperbola of the CircumTangential triangle whose asymptotes are parallel to those of the Jerabek hyperbola. It passes through X(3), X(74), X(2574), X(2575), X(3098), X(3579), X(7689), X(7691).

K905 is its inverse in the circumcircle (O) of ABC hence it is a strophoid with node X(3) and nodal tangents parallel to the asymptotes of the Jerabek hyperbola.

The singular focus is X(74) and its real asymptote is the parallel at X(110) to the Euler line.

K905 is the reflection of K725 about X(3) and the homothetic of K038 under h(X3, –2).

K905 and K001 share the same points at infinity and must meet at six other finite points which also lie on the rectangular hyperbola (h) passing through X(3) counted twice, X(74) and the points Va, Vb, Vc.

A simple construction :

Let S be a variable point on the Euler line. The line X(74)-S and the circle C(S, X3) meet at two points on K905.