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Liberal Kansas is close to the halfway point.
Kansas
Otter Village, IN
According to MeetWays, halfway between Cheyenne, Wyoming and Wichita, Kansas would be a point near Brewster, Kansas on I70.
Kansas City, Missouri.
Salina Kansas
Green Castle, Missouri.
The halfway point between Denton, Texas and Fargo, North Dakota on US75 is a point just south of the Nebraska/Kansas border, north of Sabetha, Kansas.
Westfall, Kansas.
Rishel, Kansas.
Terre Haute, IN
The halfway point is basically the area where Nebraska, Colorado and Wyoming meet. This is not an exact answer, but in relative terms this is the halfway point between Salmon, Idaho and Kansas City, Missouri. If your looking for a town, then I would say the capitol city of Wyoming (Cheyenne) is pretty much as close as your going to get to a city halfway between Salmon and K.C.