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X(1), X(4), X(8), X(10), X(40), X(65), X(72), X(3176), X(5930)

vertices of extouch triangle

vertices of the outer Garcia triangle (see ETC, preamble of X5587)

X1-OAP points, see K004, K806, K807 and also Table 53

their reflections about X(10)

The Spieker cubic is the central pK with center X(10), pivot X(8), pole X(37). Its three real asymptotes are the parallels at X(10) to the internal bisectors of ABC. It meets the circumcircle at the same points as pK(X6, X72). The inflexional tangent is the line X(10)X(37), the tangent at X(10) to the Kiepert hyperbola.

It is anharmonically equivalent to the Thomson cubic. See Table 21 and also K707.

The isogonal transform of K033 is K318 = pK(X1333, X21) and its isotomic transform is pK(X274, X314).

Locus properties :

  1. locus of point M such that the parallels at M to AI, BI, CI meet the corresponding sidelines of ABC in three points forming a triangle perspective to ABC. The locus of the perspector is a remarkable isotomic pK with pivot X(75) called Spieker perspector cubic or K034.
  2. locus of point M such that the polar lines of M in the three excircles bound a triangle perspective to ABC (together with the line at infinity and the radical circle of the excircles) (Jean-Pierre Ehrmann, Hyacinthos #9605)
  3. denote by A'B'C' the extouch triangle (cevian triangle of X(8), pedal triangle of X(40)). The line through M and A' meets the A-excircle again at A*. Define B* and C* similarly. The triangles ABC and A*B*C* are perspective if and only if M lies on K033 (together with the line at infinity and the conic through the traces of the centroid and the Nagel point whose center is the Spieker center X(10)) (Paul Yiu, Hyacinthos #9607).
  4. denote by PaPbPc the pedal triangle of point P. The internal bisector at P in triangle PPbPc meets the line BC at A'. Define B' and C' similarly. The triangles ABC and A'B'C' are perspective if and only if M lies on K033 (Paul Yiu, Hyacinthos #1947).
  5. locus of intersections of any circum-conic through X(8) with the polar of X(65) in this conic.
  6. the cevian (resp. anticevian) triangle of any point on K366 (resp. K345) is perspective with the outer Garcia triangle and the perspector lies on K033. When "outer" is replaced with "inner", we find the cubic K679.
  7. locus of M whose cevian or anticevian triangle is perspective with the Extangents triangle (see TCCT, 6.17). In both cases, the perspector lies on K750.
  8. for any point M on K033 [resp. K964] the anticevian triangle of M is perspective to the 2nd [resp. 3rd] extouch triangle and the locus of the perspector is K880 = pK(X37, X329) [resp. K963 = pK(X1427, X5932)]. See ETC, preamble of X(8856).