What is the approximate maximum separation measured along the surface of the Earth between two stations communicating by EME?

Earth-Moon-Earth (EME), or moonbounce, relies on using the moon as a passive reflector for radio signals. For two stations to communicate using this method, they both must have a geometric line-of-sight to the moon at the exact same time. The absolute maximum distance between two points on Earth that can simultaneously see the same object in space is roughly half of the Earth's circumference. The Earth's circumference is about 24,000 miles, making the maximum possible separation approximately 12,000 miles. While the terms 'perigee' (moon's closest point to Earth) and 'apogee' (furthest point) dictate path loss and signal strength, they do not change the basic geometry of the Earth. The terrestrial distance limit is strictly defined by the curvature of the Earth and mutual visibility.