You can find the vertex equidistant from other vertices with spheres.
Same model in conceptual mode (with x-ray on).
Intersecting spheres gives you this:
(same)
Here, I have used 3point UCS to align construction plane with the upper vertex and two side vertices. This allows me to make an lwpolyline in the shape of a triangle where I want it.
I've returned to the world UCS so I can polar array that triangle.
Copy three triangles and use ALIGN.
You can continue with two or more new triangles.
If something winds up "flipped backwards", MIRROR3D can help.
(the result)
This can get visually confusing. What's in front? What's in back? Placing an arbitrary solid (here a sphere is used) in the middle of everything can help.

Once the wireframe is completed. You could encompass everything with a big solid, and then use the triangles to SLICE the big solid in umpteen locations. However, I have a routine called pslice (located in P:\CADD\Instructor Folders\tbrabant\3dvis\platonics) that can automate it. It can be APPLOADed, and will introduce a new command called SLICIT.
For best results, use it in a drawing containing little else other than the triangles.