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K795

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X(2), X(3), X(110), X(111), X(1340), X(1341), X(2782), X(5968), X(9159), X(9828)

The Jerabek hyperbola (JT) of the Thomson triangle contains the ETC centers X(i) for i = 2, 3, 6, 110, 154, 354, 392, 1201, 2574, 2575, 3167, 5544, 5638, 5639, 5643, 5644, 5645, 5646, 5648, 5652, 5653, 5654, 5655, 5656, 5888, 6030, 7712, 9716.

Its Psi transform is the strophoid K795. See the analogous strophoids K509, K794 and K796 where a generalization is given.

The singular focus is X(3) and the real asymptote passes through X(262), X(381), X(671), X(2782) and X(5968) which is its intercept with K795.

Note that the focal tangent at X(3) passes through X(691), X(842) and X(5968).

The polar conic (C) of X(3) is the circle which also contains X(2) and X(182).

The nodal tangents at X(2) are parallel to the asymptotes of the rectangular circum-hyperbola passing through X(1316).

K795a

Let (P) be the parabola with focus X(3) and directrix the orthic line of the strophoid, namely the line parallel at X(2) to the asymptote.

For any T on (P), the reflection of X(2) about the tangent at T to (P) is a point M on K795.

Since the perpendicular bisector of GM is the tangent at T to (P), the parabola (P) must be tangent to the nodal tangents above.

The circle with center T passing through G is tangent at M to K795.

The Psi transform of this circle is the tangent at Psi(M) to (JT).

K795b

A simple construction

Let P be a variable point on (O).

The parallels at G to the lines PX(98), PX(99) meet the line OP at M1, M2 on K795.

These points are harmonic conjugates with respect to O and the point Q, the second intersection of OP and the polar conic (C) of O in K795.

The midpoint S of M1, M2 lies on the orthic line (L) of the cubic which is the tangent at G to (C).