Ok, I actually think I understand after reading your post a few times. Instead of including a file based on the value of "page", you'd rather just have one file included and just show content from that page based on "content."
Does your include file contain html? or does it include php code to be parsed? The reason I ask is because if your include file contains just plain html, include() will just send it to the buffer to be sent to your browser (php can do nothing at this point.) If that's the case, you'll need to use something like fread() to read the contents of the file into a php variable and then parse out the section you want.
Another way to do this is to make your include file a php file with code to parse. For example, you could create an array on that page with each section as an array element with a corresponding associative key.
$content = array(
'mission'=>"the mission of my thing is to...",
'other'=>"here's an additional section for example...",
);
Then from the main page you can call the content to show like this:
echo $content[$_GET['content']];
