Check out the eyeballing game. You're asked to "eyeball" various geometric constructions -- the perpendicular to a given segment, the center of a given circle, the bisector of a given angle, and so on.
I'm kind of curious which sort of mathematicians do best on this. I have very little reason to draw geometrically accurate pictures in my work. Although to be honest, even people who do geometry don't have to draw really accurate pictures; I suspect it's the engineers who would be best at this.