Pompeiu's theorem and the moduli space of triangles

We introduce a kind of converse of Pompeiu’s theorem. Fix an equilateral triangle $\mathrm{△}{A}_0{B}_0{C}_0$, then for any triangle $\mathrm{△}ABC$ there is a unique point $P$ inside the circumcircle ${\mathrm{\Gamma }}_0$ of $\mathrm{△}{A}_0{B}_0{C}_0$ such that a triangle with edge lengths $P{A}_0,P{B}_0$, and $P{C}_0$ is similar to $\mathrm{△}ABC$. It follows that an open disc inside ${\mathrm{\Gamma }}_0$ 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.