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K810

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X(2), X(3), X(4), X(3426)

infinite points of K243

Q1, Q2, Q3 vertices of the Thomson triangle

their images U', V', W' under h(X3, 3), see K004

foci of the inellipse with center X(3)

X3-OAP points, see Table 53

The X3-OAP points are the base points P1, P2, P3, P4 of a pencil of rectangular hyperbolas generated by :

• (H1) passing through X3, X5, X155, X576, X2574, X2575 hence homothetic to the Jerabek hyperbola,

• (H2) passing through X487, X488, X2043, X2044, X3413, X3414 hence homothetic to the Kiepert hyperbola.

This pencil defines an isogonal conjugation with respect to the diagonal triangle (D) of P1P2P3P4 : the image M' of a point M is the intersection of the polar lines of M in the two hyperbolas above. Note that the circumcircle of (D) is C(X140, 3R/4).

It follows that the locus of M such that M, M' and a fixed point Q are collinear is an isogonal pK with respect to (D).

There is one and only one such pK which is also a circum-cubic in ABC and this is K810 obtained when Q = X(3).

K810 is spK(X376, X3) as in CL055. Its isogonal transform is K851.

***

K810 belongs to the pencils of circum-cubics generated by :

K003, K187, K376, K443, the isogonal transform of K810, all passing through X(3), X(4), the foci of the inellipse with center X(3),

K002, K615, K581, K759, the union of (O) and the Euler line, all passing through X(2), X(3), X(4), the vertices Q1, Q2, Q3 of the Thomson triangle,

K006, K026, K405, K762, K811, all passing through X(3), X(4), the X3-OAP points,

K243, the union of the Jerabek hyperbola and the line at infinity, all passing through X(3), X(4), X(3426),

K004, K028, all passing through X(3), X(4) (double), the points U', V', W',

K361, K426, pK(X6, X3523).

K047, K309, K447.

Generalization :

Let P(k) on the Euler line such that OP(k) = k OG (vectors) and let pK(k) be the cubic of the Euler pencil with pivot P(k) hence passing through X(3), X(4).

K810 and pK(k) generate a pencil of spKs which generally contains :

• a stelloid S(k) with asymptotes parallel to those of K003, with radial center X(k) on the Euler line such that OX(k) = (5 k + 1) / (3k + 3) OG.

S(k) is a central stelloid if and only if k = -1 (giving the union of the Jerabek hyperbola and the line at infinity), k = 3, -3/2, -3/5 giving K026, K525, K080 respectively.

• a circular cubic C(k) with singular focus F(k) on the line X3-X110 such that OF(k) = (k - 1) / (k + 1) OX110.

C(k) is a focal cubic if and only if k = 0 (giving K187), k= ±1 (giving two decomposed cubics).

• two psKs namely psK1 = psK(Ω1, Q, X3) and psK2 = psK(Ω2, X2, X3) where :

– Ω1 and Q are the isogonal and isotomic transforms of P( (k + 3) / (3k + 1) ) respectively,

– Ω2 is a point on the trilinear polar of X(112), passing through X(6, 25, 51, 154, 159, 161, 184, 206, 232, 1194, 1474, 1495, 1660, 1843, 1915, 1971, 1974, 2194, 2203, 2299, 2393, 2445, 3192).

• a CircumNormal cubic as in Table 25.