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a^4 y z (y + z) + 2 ( y z) ( b^2 c^2 x) = 0

T1, T2, T3 : vertices of the CircumTangential triangle

Ka, Kb, Kc : vertices of the tangential triangle

S1, S2, S3 : vertices of the Stammler triangle

points Ua*, Ub*, Uc* as in page K001 and Table 18

infinite points of (K) = nK0(X6, X7735)

A' = 0 : a^2 + 2 b^2 : – a^2 – 2 c^2, B' and C' likewise

The locus of poles of pKs passing through the vertices of the CircumTangential triangle is K402. The loci of their pivots and isopivots are K403 and K902 respectively. See also Table 25.

K902 is the isogonal transform of K142, locus of pivots of all pKs having the same asymptotic directions as K024.

The tangents at Ka, Kb, Kc concur at X(669) hence K902 is a psK with respect to KaKbKc with pseudo-pivot X(6), pseudo-isopivot X(669).

The Stammler triangle S1S2S3 is the image of the CircumNormal and CircumTangential triangles under the homotheties h(X3, 2) and h(X3, –2) respectively. The tangents at S1, S2, S3 to K902 concur at X(523) hence K902 is a psK with respect to S1S2S3 with pseudo-pivot X(3), pseudo-isopivot X(523).

Recall that the six points S1, S2, S3, Ua*, Ub*, Uc* lie on the circle C(X3, 2R).

K902 meets the line at infinity at the same points as nK0(X6, X7735) and their six remaining common points lie on the circum-conic (C5052) with perspector X(5052), passing through X(110). X(5052) is the reflection of X(39) in X(6), on the Brocard axis.


K902 is a member of the pencil generated by nK0+(X32, X2) and the decomposed cubic which is the union of the Steiner ellipse and the Lemoine axis.

This pencil contains one circular cubic and one equilateral cubic, both not very interesting.


K902 and K405 have nine identified common points namely A, B, C, Ka, Kb, Kc, Ua*, Ub*, Uc* hence they generate a pencil of cubics passing through these same points.

This pencil contains K922 = pK(X32, X7712), passing through X(6), which is the isogonal transform of K092 = pK(X2, X11057).


(C523) is the polar conic of X(523) in K902. It is a rectangular hyperbola with center X(1302) passing through X(3), X(381), S1, S2, S3.

The orthic line of K902 is the perpendicular bisector of X(3)X(23), meeting the Euler line at X(7575) whose polar conic is also a rectangular hyperbola.