Pompeiu's theorem and the moduli space of triangles

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.