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X(3), X(32), X(3098), X(24309), X(52015), X(52016), X(52017), X(52018), X(52019)

A', B', C' : extraversions of X(24309)

infinite points of K006

vertices of the CircumTangential triangle T1T2T3

vertices of the Grebe triangle G1G2G3

see points below

Geometric properties :

K1291 is a CircumTangential cubic of Table 25. The remaining points on (O) are the vertices of the Grebe triangle, see K102.

Its isogonal transform with respect to the Grebe triangle is K1292, a stelloid, a member of the pencil generated by K643 and K1161.

Note that X(24309), A', B', C' are these isogonal transforms of the in/excenters of ABC.

Centers on K1291

P0 = X(52016) = a^2 (-a^2+b^2+c^2) (a^4+a^2 b^2+a^2 c^2-2 b^2 c^2) : : , SEARCH = 2.63040677926671, on lines {X6,X1196}, {X66,X542}, {X69,X184}, {X110,X193}, {X140,X141}, {X155,X159}.

P1 = X(52015) = a (a^4-a^3 b+a^2 b^2-a b^3-a^3 c+2 a b^2 c-b^3 c+a^2 c^2+2 a b c^2+2 b^2 c^2-a c^3-b c^3) : : , SEARCH = 1.59748809997818, on lines {X1,X41}, {X3,X142}, {X32,X1279}, {X58,X2191}, {X85,X927}.

P2 = X(52017) = a (a^6+a^5 b-a^2 b^4-a b^5+a^5 c+3 a^3 b^2 c+3 a^2 b^3 c+b^5 c+3 a^3 b c^2-8 a^2 b^2 c^2+3 a b^3 c^2+3 a^2 b c^3+3 a b^2 c^3-2 b^3 c^3-a^2 c^4-a c^5+b c^5) : : , SEARCH = 0.697625890668104, on lines {X10,X56}, {X106,X614}, {X517,X3098}.