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X(1), X(6), X(20), X(194), X(35237), X(46264)

excenters

infinite points of K003

vertices of the Grebe triangle G1G2G3

other points below

Geometric properties :

K1292 is the isogonal transform with respect to the Grebe triangle of K1291.

It is a member of the pencil of K+ generated by K643 and K1161 which also contains the union of the line at infinity and the rectangular hyperbola passing through {X6, X110, X182, X206, X1176, X1386, X2574, X2575}. This is the isogonal transform with respect to the Grebe triangle of the Euler line of ABC, hence it also passes through G1, G2, G3.

K1292 is a stelloid with asymptotes parallel to those of K003 and radial center X, homothetic image of X(550) under h(X6, 2/3) and X(182) under h(X20, 2/3).

K1292 and K003 meet again at two points on the rectangular diagonal hyperbola with equation ∑ b^2 c^2 (b^4 - c^4) x^2 = 0, which passes through {X1, X31, X1342, X1343, X2172}.

K1292 meets (O) at the vertices G1, G2, G3 of the Grebe triangle and at Q1, Q2, Q3 on the cubic nK0(X6, X7735). The antipodes on (O) of these latter points lie on K006.

Points on the bisectors of ABC

The remaining points of K1292 on the internal (resp. external) bisectors are the vertices of two triangles T1 = A1B1C1 (resp. T2 = A2B2C2) given by :

A1 = a^4+b^4-2 a^2 b c+2 b^3 c+2 b^2 c^2+2 b c^3+c^4 : -2 b (b+c) (b^2+c^2) : -2 c (b+c) (b^2+c^2),

A2 = a^4+b^4+2 a^2 b c-2 b^3 c+2 b^2 c^2-2 b c^3+c^4 : -2 b (b-c) (b^2+c^2) : -2 c (-b+c) (b^2+c^2).

T1 and T2 are both perspective to the antipedal A'B'C' of X(1352) with same perspector X(46264), the isogonal transform with respect to the Grebe triangle of X(32).

T1, T2 are both orthologic to ABC with first center of orthology X(46264) for both triangles and second centers O1, O2 respectively.

O1 = (a^4+2 a^3 b+2 a b^3+b^4-c^4) (a^4-b^4+2 a^3 c+2 a c^3+c^4) : : , SEARCH = -0.899113156078731.

O2 = (a^4-2 a^3 b-2 a b^3+b^4-c^4) (a^4-b^4-2 a^3 c-2 a c^3+c^4) : : , SEARCH = 1.72177166648457.

The isotomic conjugates of these points are very simple, namely :

tO1 = a^4-b^4-2 b^3 c-2 b c^3-c^4 : : , SEARCH = 5.21935377331647,

tO2 = a^4-b^4+2 b^3 c+2 b c^3-c^4 : : , SEARCH = 2.91809554645716.

Note that T1 (resp. T2) is orthologic to the cevian triangle of every point M on pK(X2, tO1) passing through X(2), X(7), X(8), (resp.pK(X2, tO2)).

In particular, they are orthologic to the medial triangle with first center of orthology X(46264) for both triangles and second centers the complements of O1, O2 respectively.