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X(2), X(3), X(4), X(6), X(111), X(112), X(115), X(542), X(895), X(5622), X(35902), X(48945), X(48946), X(49141)

foci of the orthic inconic

X(48945) = Psi2(X895) on lines {2,6}, {115,895}

Geometric properties :

K1276 is a focal cubic with singular focus O, whose polar conic is the circle passing through X(3), X(5), X(6), X(115).

The orthic line is the Fermat axis and the real asymptote is its image under the homothety h(O, 2). K1276 meets this asymptote at X on the line {111,112}.

X = a^2 (a^8-2 a^6 b^2-4 a^4 b^4+2 a^2 b^6+3 b^8-2 a^6 c^2+13 a^4 b^2 c^2-3 a^2 b^4 c^2-12 b^6 c^2-4 a^4 c^4-3 a^2 b^2 c^4+18 b^4 c^4+2 a^2 c^6-12 b^2 c^6+3 c^8) : : , SEARCH = 0.879702656232779.

Locus properties

• K1276 is the locus of point M such that X(115), M, Psi2(M) are collinear, where Psi2 is the transformation described in pages K018 and K1142. Note that K018 is the locus of point M such that G, M, Psi2(M) are collinear.

• K1276 is the locus of contacts of tangents drawn through O to the circles passing through X(6) and X(115).

• K1276 is the locus of points from which the oriented segments [G, K] and [X115, X895] are seen under equal angles.

***

K1276 and K1142 are both invariant under Psi2 and pass through X(2), X(4), X(6), X(111), X(112), X(5622), X(35902) and the circular points at infinity. K018 and K1143 are the only invariant circum-cubics.

K1276 and K1142 generate a pencil of circular cubics which is stable under Psi2. This contains two decomposed cubics, namely :

• the union of the line (L1) through {4, 6, 35092} and the circle (C1) through {2, 111, 112, 5622},

• the union of the line (L2) through {6, 112, 5622} and the circle (C2) through {2, 4, 111, 35902}.

Note that each line or circle is the Psi2-image of the other line or circle.

One of the most remarkable members of this pencil is K1277.

***

K1276 and K002 pass through X(2), X(3), X(4), X(6) counted twice, and two pairs of G-Ceva conjugate points on two lines passing through K and parallel to the asymptotes of the circum-conic with perspector X(187), passing through K and X(110).