You can create a vertex in the middle of the plane poking the plane's face (AltP).
Then set the snap type to vertex (CtrlShiftTab -> Vertex) and extrude vertices (select + E) to the plane's middle vertex while holding Ctrl so you have snapping temporarily enabled while dragging.
If it is a single object, its even easier - select each 2 vertices and hit F to make a connection (instead of extruding).
I hope this is what you are trying to achieve.
UPDATE (after the question was modified):
So you didn't want to extrude anything, you just need to split an edge.
Then there are several tools for such task - namely Subdivide (W -> Subdivide), Knife tool K and Loop cuts Ctrl + R).
Just fiddle with these tools on some testing scene to see what will work for you best.