MathNet 2019-part-02 — 184 problems (EN–VI) Question 179

Let $A, B, C$ and $P$ be four points of the plane such that $ABC$ is an equilateral triangle and $AP \lt BP \lt CP$. Suppose that knowing only the lengths $AP, BP$ and $CP$ allows us to determine the length $AB$ uniquely.

Prove that $P$ lies on the circumcircle of $ABC$.