John Gill Commentary Joshua 12:2

"Sihon king of the Amorites, who dwelt in Heshbon, and ruled from Aroer, which is on the edge of the valley of the Arnon, and [the city that is in] the middle of the valley, and half Gilead, even unto the river Jabbok, the border of the children of Ammon;" — Joshua 12:2 (ASV)

Sihon king of the Amorites, who dwelt in Heshbon
Which he took from the Moabites, and made his capital city, (Numbers 21:26) ;

[and] ruled from Aroer, which [is] upon the bank of the river of Arnon ;
a city of Moab, which never fell into the hands of Sihon, and therefore he is said to rule from it but not over it:

and from the middle of the river ;
that is, the river Arnon, which being the boundary of the Moabites and Amorites, the king of the Amorites might be said to rule from the middle of it:

and from half Gilead even unto the river Jabbok, [which is] the border
of the children of Ammon ;
so it is said to be, (Deuteronomy 3:16) ; it should be rendered, not "from half Gilead", but "and half Gilead", as it is in the Hebrew text, and so in the Targum; for half Gilead belonged to the kingdom of Sihon, as the other half did to the kingdom of Og, as in (Joshua 12:5) ; and so Jarchi remarks.