Roundabout island at a three-way junction

Centres of a Triangle

The engineer laying out this junction has three separate demands: make the island as large as possible and have it meet all three roads; put the lighting mast where it sees all three corners from the same distance; carry the platform on a single column without tipping. All three sound like 'put it in the middle'. But there is no single middle — the three demands hand back three different points, and the engineer gets to choose none of them.