arcpy - How to add vertex on the center of a polyline object?

I've got an arcpy.Polyline() object and I have to insert a new vertex in it, which has to be placed on a center of a line. Right now I can calculate the center point coordinates but how to add it to an existing Polyline object in the right place of points order? The existing vertices order should not be changed, so the output line has to have previous vertices plus the new one somehere in the middle.

with arcpy.da.UpdateCursor(line_in,'SHAPE@') as uc:
    for row in uc:
        Midpoint = row[0].positionAlongLine(0.50,True).firstPoint   

        mid_p = arcpy.Point(Midpoint.X,Midpoint.Y)

ArcGIS 10.2 has no segmentAlongLine method so is it possible to update line's vertices here?

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Due to the suggestion that this question has been already asked I would like to clarify that I don't need a new layer with points. The goal is to add a vertex to existing line.

Answer

Recently with the piece of advice of Sergei Norin I have found the solution for this case. It is about calculations of distances between vertexes within the line.

with arcpy.da.SearchCursor(line_in,'SHAPE@') as sc:

    for row in sc:

        aa = list(row[0].getPart(0)) # GETTING A LIST OF VERTEXES AS POINTS


        fst = arcpy.Array() # THE ARRAY OF COORDINATES OF THE FIRST HALF OF THE LINE
        snd = arcpy.Array() # THE ARRAY OF COORDINATES OF THE SECOND HALF OF THE LINE

        midpoint = row[0].positionAlongLine(0.50,True).firstPoint   
        mid_p = arcpy.Point(midpoint.X,midpoint.Y) #CREATING THE MIDPOINT OF THE INITIAL LINE

        snd.add(mid_p) # ADDING THE MIDPOINT AS A DEFAULT START POINT IN THE SECOND HALF


        mediana = row[0].getLength()/2 #GETTING THE HALF OF THE INITIAL LINE'S LENGTH

        nu = 0 # THE LENGTH CALCULATING FROM EACH VERTEX FROM THE BEGINNIG OF A LINE
        for i, p in enumerate(aa): #LOOP THROUGH VERTEXES' 
            try:
                ini = i # GETTING THE NUMBER OF INITIAL LOOPING VERTEX
                nex = i+1 # GETTING THE NUMBER OF THE NEXT VERTEX

                # GETTING THEIR COORDINATES


                px0 = aa[ini].X 
                px1 = aa[nex].X
                py0 = aa[ini].Y 
                py1 = aa[nex].Y

                # CALCULATING A DISTANCE BETWEEN INITIAL AND NEXT VERTEXES

                dif = math.sqrt(math.pow((px1-px0),2)+math.pow((py1-py0),2))


                # ADDING VERTEX TO A NEW  COORDIANTE LIST

                poi = arcpy.Point(p.X,p.Y) 
                fst.add(poi)



                nu=nu+dif # SUM DISTANCE BETWEEN VERTEXES WITH A TOTAL LENGTH FROM THE BEGINNIG

                #IN CASE WHEN THE LENGTH OF PREVIOUSLY COUNTED LENGTH IS MORE THAN "MEDIANA" PARAMETER, THIS LOOP ENDS UP WITH ADDING THE REMAINING POINTS TO THE LIST


                if nu>mediana:
                    fst.add(mid_p)
                    for n, j in enumerate(aa):
                        try:

                            px3 = aa[nex].X
                            px4 = aa[nex].Y
                            pei = arcpy.Point(px3,px4)
                            fst.add(pei)

                            nex=nex+1
                        except:
                            pass    


                    #fst.add(mid_p)

                    break
            except:
                    pass



# A NEW LINE IS CREATED AND PLACED 

f_lin = arcpy.Polyline(fst,sr)

icc = arcpy.da.InsertCursor(line_in,'SHAPE@')
icc.insertRow([f_lin])

del icc