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K1116

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X(3), X(6), X(23), X(110), X(511), X(10620), X(33900)

vertices of these triangles :

• Thomson (points Qi)

• tangential of Thomson (points Ri)

Geometric properties :

K1116 is a member of the pencil of cubics generated by K172 and K1114. This is studied in Table 71.

It is the only circular cubic of the pencil with singular focus F, the inverse of X(20126) in (O), on the lines {3,67}, {23,94}.

The inversive image of K1116 in (O) is a circular cubic (blue curve) passing through X(2), X(3), X(30), X(110), X(187), the vertices of the Thomson triangle. Its singular focus is X(1511), the midpoint of X(3), X(110). This is a member of the pencil mentioned at K913.

These two cubics have 6 common points on (O) and three other points on the line passing through X(3), X(373), X(3066), etc, one of them being X(3).

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Other related cubics

The circular cubics passing through the vertices of the Thomson triangle and the tangential triangle of the Thomson triangle pass through O and form a pencil.

The line passing through O and parallel to the real asymptote of one of these cubics meets the cubic again at a point of K1118.

This pencil contains K1116 and K1126, K1127, K1128, K1130.