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X(6), X(7), X(37), X(142), X(1400), X(1418), X(1439), X(1486), X(1827), X(2293), X(3668), X(13476), X(15185), X(15320), X(22277)

A', B', C' : vertices of the intouch triangle

other points below

Geometric properties :

K1089 is the SS{a → √a} image of K1088.

(S) is the circum-conic with perspector the barycentric product P* = X(42) x X(142), the isopivot of K1089. It meets K1088 at S1, S2, S3 such that the incircle is inscribed in the triangle S1S32S3. P* lies on the lines {6,692}, {7,1002}, {42,181}, {51,2187}, {55,579}, {65,1439}, {142,354}, etc.

The contacts R1, R2, R3 of the incircle with the triangle S1S32S3 also lie on K1088.

K1089 meets the line at infinity at the same points as K1090 = pK(X213, X9). The six remaining common points are A, B, C, X(6), X(37), X(1400) on the circum-hyperbola (C) with perspector X(512).