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X(4), X(20), X(98), X(468), X(1503)

X20-OAP points. See Table 53

The X20-OAP points are X(4) and the images Ha, Hb, Hc of the vertices of the orthic triangle A'B'C' under h(X4, 3) hence they lie on the altitudes of ABC.

They also lie :

• on the circular cubic K824 whose singular focus F = X(14689) is the midpoint of X(20), X(112).

• on the stelloid (S) passing through X(4), X(847), the infinite points of K003 and whose radial center X is the common point of the lines 2,389 - 30,52 - 51,381 - 376,511 - 378,576 - etc. X = X(14831) in ETC.


K824 is an inversible cubic under the inversion with center H that swaps X(20) and X(468).

The circle of inversion (C) is the homothetic image of the polar circle (P) under h(X4, √3). These circles are real if and only if ABC is obtusangle as shown in the figure.

Note that this inversion swaps ABC and HaHbHc. It maps X(98) to X(14900), a point on K824, on the lines 4,98 - 20,648.