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X(2), X(3), X(382), X(627), X(628), X(1991), X(3095), X(3413), X(3414)

X(22113), X(22114) anticomplements of X(627), X(628) resp.

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

(contributed by Richard Hilton)

Let each side of a triangle be rotated about each of its end-vertices, by equal angles θ, externally if θ is positive. Next let the outer ends be joined for each pair of segments that share a common vertex. Then the perpendicular bisectors La, Lb, Lc of these joins are concurrent, at P(θ), as in the diagram.

Adding π to θ is equivalent to reflecting each of the joining segments in the corresponding vertex of the reference triangle, so that the mid-point M(θ) of P(θ) and P(θ+π) is the point of concurrency of the lines through the vertices that are perpendicular to the joining segments.

This point M(θ) lies on the Kiepert hyperbola. Also P(θ) and P(θ+π) are always collinear with the centroid G.

The locus of P(θ) as θ varies is the quartic Q088 with two pairs of parallel asymptotes, each pair having an asymptote of the Kiepert hyperbola as mid-line. The locus is shown above, together with the Kiepert hyperbola and asymptotes.

Note that G is an acnodal point on the curve.

Additional properties

• The line M(θ)M(-θ) contains K and bisects the parallel lines P(θ)P(-θ), P(π+θ)P(π-θ).

• The line P(θ)P(-θ) envelopes a hyperbola (H) with center K passing through X(511), X(524), the midpoints of X(3)X(141), X(69)X(182), X(193)X(576). This hyperbola is quadritangent to Q088.

• The lines P(θ)P(π-θ) and P(-θ)P(π+θ) meet at S(θ). The locus of S(θ) is a hyperbola (H') passing through X(2), X(39), X(99), X(141), X(550), X(2896), X(3098), X(3413), X(3414).

• The midpoints of P(θ)P(-θ) and P(π+θ)P(π-θ) lie on the line through G and S(θ).