In Plane Sight

Flatland is in tumult after a spate of triangle-on-triangle violence, and the prevailing attitude is one of utter paranoia. The triangles, which can see out of each of their three vertices, are so mistrustful that they will not tolerate the presence another triangle unless they can see each of the other's vertices, and, moreover, see it with each of their own eyes. Triangles are assumed to be disjoint and opaque.

It is your job to help as many triangles as possible congregate, in a fashion acceptable to all participants, to promote dialogue and peace. Find yourself a pen, paper, and a bag of Doritos, and consider:

Problem: Can three equilateral triangles of equal size be placed in a 'acceptable' arrangement? (If yes, what is the upper limit, if any?)


Notes:

-I do know the answer, but it took me awhile. I've never really understood triangles...

-I haven't read Abbott's 'Flatland' since maybe age 12, so I have no idea how faithful this scenario is, or whether related visibility issues were explored in the book. As far as I know this puzzle has not been posed elsewhere, but I do recall there being at least one interesting visibility puzzle in Engel's book 'Problem Solving Strategies'.

-Notice how I refrained from saying 'acute paranoia'... it wasn't easy, but I think I'm a better person for having done so.