I know you said the delauny wouldnt work but i think its a pretty good approach. The circle packing algorithm essentially does what you are looking for but for one point only which i guess you could extend to three points or triangles.
With the delauny algorithm you make a mesh and providing that you can capture the vertices of each triangle you could then move them some random amount towards its own centre. This way you dont just get triangles with equidistant vertices to its neighbours.
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