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X(2), X(110), X(111), X(5108), X(5980), X(5981), X(9828), X(9829), X(9830), X(9832)

points described below

(contributed by Peter Moses, 2016-04-13)

The Jerabek hyperbola (JG) of the Grebe triangle (see K102) contains the ETC centers X(i) for i = 2, 3, 6, 83, 99, 194, 3413, 3414, 5652, 6033.

Its Psi transform is the strophoid K796. See the analogous strophoids K509, K794, K795.

The nodal tangents at G are the axes of the Steiner ellipses.

The real infinite point is that of the lines X(2)X(353), X(6)X(671), etc.

The singular focus F is the second intersection of the line X(2)X(187) with the circle (C) passing through X(3), X(110), X(111), X(187), X(351), X(2482) with center X(9126), the midpoint of X(3), X(351). F has SEARCH = 6.16722156913829.

The line X(3)X(76) meets K796 at two points inverse in (O) namely X(5980), X(5981) and a third point S lying on (C) with SEARCH = 4.95446782777643. S is the Psi image of X(5652), the orthocenter of X(2), X(3), X(6).

These three latter points were added to ETC as X(9830), X(9829), X(9828) on April 16, 2016. Also, X(9832) on the Euler line.

Generalization :

The Jerabek hyperbolas (JT), (JG) of the Thomson, Grebe triangles generate a pencil of rectangular hyperbolas passing through X(2), X(3), X(6), X(5652). The Psi transforms of these hyperbolas are strophoids with node X(2) and singular focus on (C). All these strophoids pass through X(2), X(110), X(111), S.