Tool for Creating Points Intersecting on Corner of Polygon

I recently used the Intersect Tool for a polyline and polygon layer where I wanted a point file for all the points of intersection of the polyline with the polygon. I even used it for polylines intersecting at polylines. Perfectly fine.

However, I noticed that for the points layer that I created with the polyline-on-polygon set, there weren't any points of intersection at the corner of polygons. The polyline didn't intersect the polygon but I still want to see a point at the location where the intersection is parallel to the border of the polygon. I've attached a picture of what I'm looking for.

I'd just like to avoid having to do this one-by-one in terms of creating points because I run the risk of the resulting points layer not being copasetic with the other points layer(s) I've created with the Intersect Tool.

Thanks again--I've benefitted immensely for the most part from this forum.

Dan B

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