Geometric "proof" by exhaustion

[Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?](http://math.stackexchange.com/q/896725/5800) This [project](https://github.com/l0b0/graphics/tree/6dcb4e254409324410553dc62b7613b56a2ec0fb/chisel) is an attempt at a layman's "proof" by exhaustion that this is not possible: each of the objects have one profile which is imperfect. The screenshots show orthogonal views of each object (yellow) overlaid with one of the othe