We have Bluecoat proxies with Inline mode.
I find from our inside network, the trace to google goes through a different path than to the proxies. The IE has configured with the URL of the proxies for HTTP traffic
I wonder how does a router in between know whether this is a HTTP traffic or not?
If you are not using WCCP on the routers and have simply configured the proxy in the web browser the router doesn't know it is http traffic.
The reason you see a different path is probably because the proxy is reachable via R9 whereas direct internet access goes to R2. When you type a URL into your browser the traffic is then sent to the proxy so the router simply routes it to the proxy address.
A traceroute from your computer is not http and is not done via the web browser so it is not sent to the proxy hence the reason it is sent direct.