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K805

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X(4), X(6), X(39), X(69), X(141), X(1350), X(14994)

infinite points of K102 or K141

A', B', C' : vertices of the pedal triangle of X(1350)

reflections of A, B, C, X(39) about X(141)

X(6)-OAP points, see Table 53

other points below

K805 is a central cubic with center X(141) and asymptotes parallel to those of the Grebe cubic K102.

It meets the circumcircle at the same points as pK(X6, Z) where Z = X(14929) = X(5), X(183) /\ X(30), X(69) /\ X(99), X(550), etc.

The reflection S = X(14994) of X(39) about X(141) lies on K805. S is also the midpoint of X(69) and X(76).

It also contains the four X(6)-OAP points i.e. the points P1, P2, P3, P4 whose AntiPedal triangle has Orthocenter X(6), see also Table 53.

K805a

These points also lie

• on the cubic (K) = pK(X4 x X1350, X4) passing through X(4), X(371), X(372), X(1350),

• on the rectangular hyperbola (H) passing through X(2), X(69), X(3413), X(3414), the reflection of X(39) about X(141). (H) is homothetic to the Kiepert hyperbola.

See also K819, K820.