When the moon is full, it always looks brightest to us on Earth vs. any other phase since we see its full illuminated area. It’s the same for Mars and the other outer planets, as we see their “full” phase when they are also closest to the Earth, i.e. when they are aligned on the same side of the sun as the earth. Not so for Venus and Mercury. Because their orbital radii are smaller than Earth’s, they present less and less lit surface to an earthbased observer as they approach their closest distance to our planet. Therefore the position at which these two planets look the brightest from Earth is at some point where their orbital radius has some nonzero angle to that of Earth’s.
Develop the relationship on how bright Venus looks from the earth as a function of the angular difference in orbital positions, and solve for the angle where Venus looks the brightest from earth. Simplifying assumptions: the orbits are perfect circles in the same plane, and the orbital radius of Venus is 0.723 that of earth (a good avg. value).
Bonus for purely analytical answer.
(In reply to
what I get by Charlie)
The 118° angle at Venus and the 22° angle at the Sun imply a 40° angle at the Earth. This angle, in astronomical terms, is called the elongation of Venus.
The Wikipedia entry on Aspects of Venus lists July 12, 2015, as the next occurrence of Venus's greatest brilliancy. An ephemeris lists the elongation of Venus for that date as 38°, within 2° of the result using the simplified assumptions.

Posted by Charlie
on 20150409 20:05:18 