Keep getting re-infected over the LAN?

Should check and make you do not have an mbr virus that stayed with you when you reinstalled winodws:
http://superuser.com/questions/643413/how-to-find-out-if-my-mbr-is-infected

Does your pc have additional hard drives in it besides the boot drive, virus could be on it.

I would scan with malwarebytes, ive seen avg miss plenty of infections or call out false postivies.

Scan the network pcs as well.

You have a trojan somewhere on the network. It needs to be eradicated.
You need to isolate every machine and eradicate it everywhere.

Okay, first off your doing a false positive. The 'infected' files noted are ones it found in the IE Cache from 'going' to sites with infected files on them. It does not mean the computer itself was infected with that. So AVG was doing it's job and you / whomever needs to stop visiting those websites. This can also occur in the background if you have a bunch of toolbars installed as well. I would (on IE) click the Gear icon top right, Options, then Advanced and run reset with the check mark to remove all personal settings, to clear out the browser.

Now second to that (boosted1g was wrong on how he said it but right on the inference) NO Antivirus will detect Malware, simply put it was never designed to do it. Running Malwarebytes regularly will help remove the malware (like the toolbars) that the AV wont' catch.

Now your talking a LAN of computers. How many are we talking? Because until the 'infected' (there is more then one) computers are also fixed EVERYONE gets reinfected with some of the versions out there, or worse the same 'person' keeps using multiple computers and does it by visiting those websites the person shouldn't be.

If we are talking a business or school, this would be a Management issue to address and potentially terminate / kick out someone whom isn't following the rules. You would also need to install a proper set of safeguards, policies and business level security to combat this or just 'deal' with the issue if everyone (management) isn't going to take it seriously until something big (WHAT do you mean our $1M Contract is gone? How can it be gone!) occurs to prove to them they need to address it properly.

Your indicating "the computer seems to be infected" but you keep saying all scans, other then a 'cache' file from browsing shows no other infection or malware. So how are you concluding the computer is in fact infected? Is the p[roblem of just unable to browse other PCs's on the network your only symptom, because that is NOT typical symptom of a infection. That is typical for other reasons (DNS not working properly, misconfiguration of WINS, no DMAIN structure in place, conflicting FW (including built in Microsoft FW) settings, etc.). To me your actually disproving yourself and contradicting yourself.

So let's address the current problem logically, is there a infection or not?
IF your only evidence is cache browser data then yes someone is trying to cover their tracks. BTW CCleaner works ONLY on the 'logged in user' profile not on 'all profiles', which each profile individually has their own IE Caches separate. So running CCleaner wouldn't' really clear ALL IE caches. Further if they used a Pen/Thumb Drive with Portable Apps, then yes a cache would be left behind, but the actually application wouldn't be detected because it actually was on the user's Pen / Thumb drive only.

What you should have in place is a Domain Structure, securing down all assets with require ID and passwords. This limits the capabilities of people messing (covering tracks) with the assets, and allows a detail log of who was on, when and what assets they accessed. Further this protects sensitive materials (i.e. ask the Managers if they like people knowing what their pay rate is, or the credit cards numbers and other data used on the recent business expense?). This would be the first research data to know whom was using (logged on) that infected PC during that time the file was created.

Next would be to setup a Proxy server forcing ALL Internet Access through it. This can tie into the Domain Account, so that you can limit say people from using Torrent or preventing access to Sluts.Com websites. Which is something you can EASILY provide for management, simply provide ANY recent case on 'Sexual Harassment in the Workplace' lawsuits about porn images being seen on Company assets. Just provide the 'compensation' cost (not even the legal fees) and see how quickly they will demand that the proxy server be installed pronto. This will provide proof positive of who access what website, and allow (for a fee) constant updated lists to be 'blacklisted' preventing such access and the cost for all the repairs being done.

Lastly the only FireWalls needed are for your LAN built into the Router connected to your ISP. It also should be using IDS (Intrusion Detection System) with the Proxy to provide specialized rule handling and 24x7 monitoring of the communications for any issues. This would also improve through the ability to 'browse PC's on the network' as the Domain (Active Directory) is managing all the communications, and thus knows all the assets that are 'browserable'.