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K927

(b – c)2 (c y – b z) y z = 0

X(1), X(514), X(1111), X(3413), X(3414), X(5997), X(5998)

isogonal conjugate of X(4251)

A', B', C' : vertices of the incentral triangle = cevian triangle of X(1)

A", B", C" : vertices of the anticevian triangle of X(514)

other points below

Geometric properties :

K927 is a remarkable cubic that passes through the common points of the incircle and the Steiner inellipse, one of them being X(5997). See below for other related points and a figure.

See the related K554 where a generalization is given and also K925.

K927 has three real asymptotes : one is the line X(1)X(514) and the others are the parallels at X(5988) to those of the Kiepert hyperbola. Note that the sixth common point of K927 and the hyperbola is the isogonal conjugate X(4251)* of X(4251).

K927a

With U = (a – b + c)(a + b – c), V = (a + b – c)(– a + b + c), W = (– a + b + c)(a – b + c), the eight following points lie on K927 :

X(5997) = a – √U : b – √V : c – √ W

X(5998) = a + √U : b + √V : c + √ W

A1 = a – √U : b + √V : c + √ W

B1 = a + √U : b – √V : c + √ W

C1 = a + √U : b + √V : c – √ W

A2 = a + √U : b – √V : c – √ W

B2 = a – √U : b + √V : c – √ W

C2 = a – √U : b – √V : c + √ W

X(5997), A1 B1, C1 lie on the incircle and the Steiner inellipse. Note that the diagonal triangle of these four points is the anticevian triangle of X(514).

X(5998), A2, B2, C2 are their respective isoconjugates under the isoconjugation with fixed point X(514).

The six following points also lie on K927 : A3 = X(5997)A1 /\ B1C1, A4 = X(5998)A2 /\ B2C2 and B3, C3, B4, C4 similarly.

K927 must meet the incircle and the Steiner inellipse each at two other points namely I1, I2 and S1, S2.

I1, I2 are the common points of the line X(1)X(7) and the circum-conic passing through X(11), X(80), X(1111), X(1146), etc.

S1, S2 are the common points of the line X(1)X(2) and the circum-conic passing through X(334), X(903), X(1086), X(1111), X(1358), etc.

***

Let M# be the X(1)-Ceva conjugate of M i.e. the perspector of the cevian triangle of X(1) and the anticevian triangle of M. For any point M on K927, this point M# also lies on K927 since its pivot is X(1).

It follows that K927 contains :

• X(514)# which is actually X(4251)* and the perspector of the triangles A'B'C' and A"B"C",

• X(5997)#, the perspector of the triangles A'B'C' and A1B1C1, also A"B"C" and A2B2C2,

• X(5998)#, the perspector of the triangles A'B'C' and A2B2C2.

More generally, K927 contains the perspector of any pair of the seven triangles mentioned above, ABC included.