Let [unparseable or potentially dangerous latex formula] be an acute triangle with ω, Ω, and [unparseable or potentially dangerous latex formula] being its incircle, circumcircle, and circumradius, respectively. Circle ω_A is tangent internally to Ω at [unparseable or potentially dangerous latex formula] and tangent externally to ω. Circle Ω_A is tangent internally to Ω at [unparseable or potentially dangerous latex formula] and tangent internally to ω. Let [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] denote the centers of ω_A and Ω_A, respectively. Define points [unparseable or potentially dangerous latex formula] analogously. Prove that
[unparseable or potentially dangerous latex formula]
with equality if and only if triangle [unparseable or potentially dangerous latex formula] is equilateral.