Hi Aurel,
Your main window should have some callback attached to it that would be exposed to you entirely as an Oxygen WinMain() in Charles' GUI scripts, or as a series of event subs similar to VB6. In any case, you will have access to either a WM_SIZE message or something that would be similar to OnSize()/OnResize()/etc event handler.
This event handler will usually give access to the new width and height of the main window as it is being resized. For example, if you're using WM_SIZE then its associated lParam will store the new width in its low word half, and the new height, in its high word half.
You should use a call to e.g. MoveWindow() or SetWindowPos() WinAPI in this event handler or Case WM_SIZE to resize your child browser window by its own hWnd to a new size proportionately to the changes in the main window's width and height, usually preserving the original aspect ratio of the child browser window.