K034 is the isotomic pK with pivot X(75) isotomic conjugate of X(1). Hence, the tangents at A, B, C are the (internal) bisectors of ABC.
It is anharmonically equivalent to the Thomson cubic. See Table 21.
Locus properties :
- locus of perspectors as seen in Spieker central cubic.
- locus of point P such that the cevian triangle of P and the intouch triangle are orthologic (intouch triangle = pedal triangle of X(1) = cevian triangle of X(7)). The two centers of orthology lie on K033 and on a central pivotal cubic with respect to the intouch triangle passing through X(1), X(4), X(65) = pivot, X(942) = center, X(950) = isopivot, having three real asymptotes perpendicular to the sidelines of ABC. See the related cubic K154 and a generalization at Table 7.
- locus of point P such that the cevian triangle of P and the excentral triangle are orthologic. The two centers of orthology lie on K033 and K004 respectively.
- Ix is one in/excenter and A'B'C' is the medial triangle. For any point M, the parallel at Ix to BC meets MA' at Ma, Mb and Mc similarly. The triangles ABC and MaMbMc are perspective if and only if M lies on the Thomson cubic. The locus of the perspector is K034. (Philippe Deleham, 16 nov. 2003)
- If A'B'C' is the cevian triangle of X then K034 is the locus of X for which 1/BA' + 1/CB' + 1/AC' = 1/A'C + 1/B'A + 1/C'B. Compare with K007 and K200.
The isogonal transform of K034 is K175.