  In 2014, Paul Yiu constructed two equilateral triangles inscribed in a
Kiepert hyperbola associated with a reference triangle. It was asserted that
each of the equilateral triangles is triply perspective to the reference
triangle, and in each case, the corresponding three perspectors are collinear.
In this note, we give a proof of his assertions. Furthermore as an analogue of
Lemoine's problem, we formulated and answered the question about how to recover
the reference triangle given a Kiepert hyperbola, one of the two Fermat points
and one vertex of the reference triangle.
