Want this question answered?
Kansas City, Missouri.
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.
Gerardeau missouri
Consider Kansas City, Missouri.
Ohio
The shortest and fastest route between those two places has a halfway point in Kansas. If you're traveling through Missouri, Prairie State Park is near halfway.
Brooklyn, Missouri.
Macomb Missouri, as best I can figure
One halfway point between Saint Louis, Missouri and Manhattan, Kansas is Sweet Springs, MO.
Green Castle, Missouri.
Concordia, Kansas is the city that is halfway.
Assuming you're taking I-44, the halfway point is near Plano, Missouri but locations in the west part of Springfield are close enough.