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X(99), X(671)

other points described below

Let us drop perpendiculars from A, B, C on the Steiner focal axis at A1, B1, C1 and on the Steiner non focal axis at A2, B2, C2.

K370 is the unique circum-cubic passing through these six points.

K370 meets the Steiner ellipse at A, B, C, X(99) (Steiner point), X(671) (its antipode on the ellipse) and its tangent at X(99) is that of the Steiner ellipse.

The third point on X(99)-X(671) is Z = X(14971), the homothetic of X(115) under h(G,1/3).

K370 meets the circumcircle at A, B, C, X(99) and two other points P1, P2 on the line Z-X(599). X(599) is the reflection of K in G.


Properties of these points A1, B1, C1, A2, B2, C2

  1. the lines A1A2, B1B2, C1C2 concur at X(115), the center of the Kiepert hyperbola. They are parallel to the cevians of X(671).
  2. the lines B1C2, B2C1 concur at A', the trace of the line GK on BC. Similarly for the points B', C'. Recall that GK is the tangent at G to the Kiepert hyperbola and that the isotomic conjugate of its infinite point is X(671).


Squared distances

Let S1 = AA1^2 + BB1^2 + CC1^2 and S2 = AA2^2 + BB2^2 + CC2^2. W is the Brocard angle, S is TWICE the area of ABC.

We have S1 = S [cotW - root(cotW^2-3)]/3 and S2 = S [cotW + root(cotW^2-3)]/3. These have the following means :

  • arithmetic : S cotW /3 = (a^2+b^2+c^2)/6 = (GA^2 + GB^2 + GC^2)/2,
  • geometric : S / root(3),
  • harmonic : S tanW.

If L and L' are two perpendicular lines through G and if we drop similar perpendiculars on L and L', we may define two analogous sums S1 and S2. It is obvious that S1 + S2 = GA^2 + GB^2 + GC^2 = (a^2 + b^2 + c^2)/3. Thus, when one of these sums is maximum, the other is minimum. A sum is minimized when L is the focal axis and maximized when L is the non focal axis.

Note that these two sums are equal when L and L' are parallel to the asymptotes of the rectangular circum-hyperbola through the Steiner point X(99). This has perspector X(230). The common value of the sums is naturally S cotW / 3.