Finally, I was able to make the WRT54G connected to eth1 in the Core
I over sited the instruction in step 5 to connect the WRT54G to the core through the LAN port not the WAN port.
After connecting the WRT54G to one of the 4 LAN ports, I can now access the WRT54G management web page (at 192.168.80.130) and ping it.
I can also ping PC1 that is connected to another LAN port on the WRT54G
Thanks Techstyle for helping me with this problem
P.S.
Now my PC1 (Windows machine) cannot connect to the Internet when it is connected both to the external network via wireless and to
the internal network (via wired connection to the WRT54G, which is wired to eth1 in the Core)
ipconfig shows:
C:\Windows\system32>ipconfig
Windows IP Configuration
Wireless LAN adapter Wireless Network Connection:
Connection-specific DNS Suffix . : Moshkovitz
Link-local IPv6 Address . . . . . : fe80::c48:de15:4b6:a34c%11
IPv4 Address. . . . . . . . . . . : 10.0.1.2
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 10.0.1.1
Ethernet adapter Local Area Connection:
Connection-specific DNS Suffix . :
Link-local IPv6 Address . . . . . : fe80::bd61:d2fb:230e:4c1e%10
IPv4 Address. . . . . . . . . . . : 192.168.80.140
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.80.131
192.168.80.1
If I disconnect the wire connection, it can access the Internet fine.