Mercury at Elongation

Back in the '70s a young amateur assumed that whenever an inferior planet reaches maximum elongation (east or west) the radius vector of the planet is perpendicular to that of Earth. Logically, it seemed, the two radius vectors form a right angle whenever the angular distance of Mercury (or Venus) from the sun, as seen from Earth, is at its greatest. The amateur thought that, to appear farthest from the sun when viewed from Earth, the radius vector of Mercury couldn't be less than 90 degrees away from Earth's, since the planet wouldn't be fully elongated to one side of the sun.

As the diagram above makes clear, this reasoning was fallacious. Elongation occurs not when the radius vectors form a right angle but an acute one. When Mercury is at elongation, its radius vector is perpendicular to a line extending from it to Earth. But that line is obviously not Earth's radius vector--the line from Earth to the sun. The separation of the two radius vectors, at elongation, is far less than the "logical" 90 degrees. It's only about half that.

Why does reality, in this case, seem to defy logic? It must be a distance effect. When the radius vector of Mercury is perpendicular to Earth's, the planet is farther away than when it is at elongation. When the planet is farther its angular distance from the sun appears less. Conversely, when it is nearer its distance appears greater. If Mercury was much farther away its angular distance from the sun would appear much smaller; ultimately the two bodies would appear to converge. So it's not surprising the angular sun-Mercury distance appears greatest when the planet is closer to Earth than it is when its radius vector is at right angles to ours.