Kx is the Lemoine point X(6) or one of its harmonic associates and A'B'C' is the medial triangle. For any point M, the parallel at Kx to BC meets MA' at Ma, Mb and Mc similarly. The triangles ABC and MaMbMc are perspective if and only if M lies on K177. The locus of the perspector is K141. See also K002, property 10.
The parallels at point P to each sideline of ABC meet the other sidelines at six points which always lie on a same conic with center S. The line PS contains X(32) if and only if P lies on K177. See alsoK002, property 12.
Locus of isopivots of pKs passing through the vertices of the Grebe triangle. The locus of pivots is K644 and the locus of the poles is pK(X251 x X32, X251). See also Table 57.