  We introduce a kind of converse of Pompeiu's theorem. Fix an equilateral
triangle $\triangle A_0B_0C_0$, then for any triangle $\triangle ABC$ there is
a unique point $P$ inside the circumcircle $\Gamma_0$ of $\triangle A_0B_0C_0$
such that a triangle with edge lengths $PA_0, PB_0$, and $PC_0$ is similar to
$\triangle ABC$. It follows that an open disc inside $\Gamma_0$ can be
considered as a moduli space of similarity classes of triangles. We show that
it is essentially equivalent to another moduli space based on a shape function
of triangles which has been used in preceding studies.
