Illustration of James Evans' theory on how Aristarchos might have reached his famous 87° half moon elongation (in History and Practice of Ancient Astronomy, p. 72).

In this illustration, the Sun's and Moon's distances S and M are expressed in Earth radii; the time t, in hours; the length of the synodic month T, in days; and the half moon's elongation θ, in degrees.
For illustrative purposes, distances are not to scale. For true scale set the sliders to t = 0.29 hrs, T = 29.53 days, and M = 60⅓ Earth radii. For Evans' theory to work, move the sliders to t = 6 hrs, and T = 30 days.
Despite Evans' assumptions (that half moons occur closer to new moons than to full moons and that Aristarchos made a purely indoor mathematical calculation to obtain his 87° angle), the fact is that quadratures (first and last quarters) occur a mean of ten minutes closer to full moons than to new moons (Meeus, Morsels IV, p. 12) and, though geocentric dichotomies (50% illumination as seen from the Earth’s centre) always precede first quarters and follow last quarters by a mean of about 18 minutes (Meeus, Morsels V, p. 22), sometimes things are so arranged that an observer on the Earth’s surface can see topocentric dichotomies (half moons) occurring closer to full moons than to new moons! Drag the point O (representing the observer's position) along the Earth's surface to see how this is so.