Your totally right, I have no clue why you want to reinvent the wheel and make it triangular. Maybe Im missing something 
An hour on xojo without Graph data types, JSON, svg, etc, etc. Works on multiple floors, with hundreds of rooms, actually, can work with thousands of points if you want to inlcude point to point directions (bed to bed, any table in the cafeteria, etc)
Sure in the end is graph theory, but you dont need to use/store actual graphs, as @mikey said, just a little linear algebra.
@MGD yes. But you can improve that with a jump point search.
With this as a base, it takes another half hour to add Weighted Terrain Costs to make the path to avoid some areas or even better, use different privileges to create paths for public, staff, restricted staff, etc. Something that looks impractical with your idea.