Unit-circle method for a distinct-distance lower bound ($|\Delta (P)|\gtrsim |P{|}^{2/3}$)

For every positive distance, draw equal-radius circles centred at the points of $P$. Their incidences between points and curves count the ordered pairs at that distance. The Szemerédi–Trotter theorem for unit circles bounds each distance class by $O(|P{|}^{4/3})$. Summing over all classes accounts for $|P|(|P|-1)$ pairs, proving $|\Delta (P)|\gtrsim |P{|}^{2/3}$.