copying the vertex will only copy the vertex, the segments will not be included. you may use refine. or you may use insert vertex.
-to use insert vertex, select first the segment edge ( line) then right click and choose insert vertex, then point out where you need a vertex.

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sounds like you need to use extrude edge. In editable poly choose an edge, hold shift then move. you will get a new polygon. That function is pretty much the most used modelling method in max. extrude edges move, scale and rotate them then with two verts selected use ctrl+alt+c to weld things together.
with this method the general idea is to extrude out edges to roughly define the forms and topology of the mesh then gradually extrude more edges to fill in the gaps. target weld edges or vertices is also very handy.

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