Hello Maodo
The 172.16.1.0 subnet is a range of addresses that are given to us by the ISP that will be used to translate the INSIDE addresses to the OUTSIDE. In other words, when the Host communicates to the outside world, the 192.168.12.1 address will be translated to 172.16.1.X when it traverses the NAT router. So from the NAT router outwards, all communication occurs with the IP address 172.16.1.X. Note that this subnet essentially “exists” on the F1/0 interface of the NAT router. In a sense, it coexists with the 192.168.23.2 IP address on that interface.
Now concerning your questions:
We could add a static route as you suggest, however, a dynamic routing protocol is preferable. Using BGP to advertise this network from the NAT router to the ISP router means that any future changes will be automatically updated. Also, dynamic routing is more elegant. An ISP may have hundreds of customers and you wouldn’t want to go into the ISP router every time there is a change to be made, every time a new user is connected or every time a client cancels their connection.
Although it is true that the vast majority of NAT implementations occur on the EDGE of the network and thus translate between public and private addresses, the use of private and public addresses on the inside and outside of NAT respectively is not a requirement. The ranges used in this example are used just for educational purposes.
It is true however that you would never use private addresses on the Internet.
I hope this has been helpful!
Laz