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I am not aware how to add the intersection point while drawing polyline. However, below is the link for sample code to create point where polyline intersect. It is a VBA code You may need to modify the code as per your requirements.
Hope this will be helpful to start with your requirement.
So you have your feature class, say featCls
IFeature feat = featCls.CreateFeature
feat.Shape = point //point is the intersection of the pplyline
That's it, you just added the point to your feature class.