Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

too complicated to be written here. Click on the link to download a text file.

X(3), X(4), X(6), X(20), X(990), X(1350), X(1766), X(41735), X(42458), X(61086), X(61087), X(61088), X(61089), X(61090), X(61091)

infinite points of the altitudes of ABC

points T1, T2, T3 (apart A, B, C) of K169 on (O) and their antipodes

other points below

Geometric properties :

The isogonal conjugation with respect to the triangle (T) = T1T2T3 transforms K169 = pK(X6, X69) into K1361. See CL076 for a generalization and other analogous cubics.

K1361 and K169 meet at X(6), X(20), T1, T2, T3 and at four points on the parallels at X(6) to the asymptotes of the Jerabek hyperbola.

K1361 is a central cubic with center O and inflexional tangent passing through X(1196) and X(1611).

The asymptotes of K1361 are those of the Darboux cubic K004 and the remaining common points of the two cubics are X(3), X(4), X(20).

K1361 is a member of the pencil of cubics generated by K004 and the union of the line at infinity (twice) with the Euler line. This pencil also contains K1362.

K1361 meets (O) at T1, T2, T3 on K169 and their antipodes S1, S2, S3 on nK0(X6, X40132).

See K169 for other properties of T1, T2, T3. Note that the inconic (C) with center X(141), perspector X(76) is also inscribed in the triangle T1T2T3.

Other points on K1361

Q1 = a (a^5+a^4 b-a b^4-b^5+a^4 c-6 a^3 b c+4 a^2 b^2 c-2 a b^3 c+3 b^4 c+4 a^2 b c^2-2 a b^2 c^2-2 b^3 c^2-2 a b c^3-2 b^2 c^3-a c^4+3 b c^4-c^5) : : , SEARCH = 8.79728746967410, on the lines {3,1279}, {4,9}, {20,1219}, {105,165}.

Q2 = a (a^5-a^4 b-a b^4+b^5-a^4 c+6 a^3 b c-4 a^2 b^2 c+2 a b^3 c-3 b^4 c-4 a^2 b c^2-2 a b^2 c^2+2 b^3 c^2+2 a b c^3+2 b^2 c^3-a c^4-3 b c^4+c^5) : : , SEARCH = 4.76743831871859, on the lines {1,7}, {3,1279}, {6,517}, {34,1697}, {38,1709}, {40,595}, {46,1471}, {55,1465}, {106,1292}, {149,2000}, {165,614}.

Q1, Q2 are symmetric about O. They are now X(61087), X(61086) in ETC.

X(61088) is the reflection of X(41735) in O.