If I have mangled the corner of a cube, say — as in the Knife tutorial — and I did NOT save the original corner point……

If I have mangled the corner of a cube, say — as in the Knife tutorial — and I did NOT save the original corner point… is there any way to reconstruct that corner by extending the existing edges to their intersect point? or is it a lost cause if you have deleted any of the original, default vertices of a solid? in SKetchup for example, one can extend new edges out into space and they will auto-align with existing nearby edges, surfaces, midpoints etc. I was somehow expecting something like this in Blender but googling has not yet revealed it. Of course it may be in a much later lesson!