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K1121

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X(3), X(6), X(5646), X(11284), X(31884)

vertices of these triangles :

• Thomson (points Qi)

• tangential of Thomson (points Ri)

other points below

Geometric properties :

K1121 is a member of the pencil of cubics generated by K172 and K1114. This is studied in Table 71.

K1121 meets the circumcircle again at T1, T2, T3 on nK0(X6, X14930). The vertices S1, S2, S3 of the tangential triangle of T1T2T3 also lie on K1121 and on the Stammler hyperbola SH.

SRH is the reflection of SH in O, see preamble of X(33537) in ETC. It is the polar conic of O in K1121 which contains R1, R2, R3. It follows that the tangents to K1121 at R1, R2, R3 concur at O hence K1121 is a pK with isopivot O and pivot X(5646) with respect to the tangential triangle of the Thomson triangle. Recall that X(5646) is the Lemoine point of the Thomson triangle, also the perspector of the Thomson triangle and the tangential triangle of the Thomson triangle.

The tangents to K1121 at Q1, Q2, Q3 concur at X(11284) hence K1121 is a pK with isopivot X(11284) and pivot O with respect to theThomson triangle.

The tangents to K1121 at S1, S2, S3 also concur but not on the cubic which is therefore a psK with respect to S1S2S3. It follows that triangles S1S2S3 and T1T2T3 are perspective but the perspector is not on the cubic.

R1, R2, R3, S1, S2, S3 are on a same (orange) conic passing through X(2930) with (orange) center the second intersection of SRH with the line X(3)X(597).