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X(2), X(3), X(6), X(25), X(32), X(66), X(206), X(1676), X(1677), X(3162)

vertices of the tangential triangle

midpoints of ABC

vertices of the Grebe triangle

K177 is the isogonal transform of K141. Its asymptotes are parallel to those of K169 = pK(X6, X69) and also to those of pK(X2, X315).

See also K442 and K517.

Locus properties :

  1. 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.
  2. 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 also K002, property 12.
  3. 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.
  4. locus of P whose anticevian triangle is perspective (at Q) to the Ara triangle (see ETC, X5594). Q lies on K174.