I present to the world my latest invention, FBChatAT. A Facebook Chat Inline Auto Translation tool for Google Chrome. (If you don't use Chrome then click here, install, and be set free.)
This extension acts/translates in two ways:
When you receive a message in Facebook Chat, the extension will automatically translate that message into your native language, then insert the translated text underneath the original message. This allows you to review/read the original messages and translation simultaneously while you chat.
Secondly.. To type in your native language and have it automatically translated out: An extra textbox is provided at the bottom left of the screen. Type into this box your text will be translated and shown in the space provided above. If/when you are happy with your outgoing message, press 'enter'. This will push the translated text into the current active chat flyout/window. Press 'enter' a second time to push the message into the chat stream.
The default native and foreign languages are English and Traditional Chinese respectively. You can set your preferred languages in the options page of the extension. (Right click FB button in toolbar and choose options).
You may find the full source for this extension on GitHub.
This extension uses Google Translate for the translation.