How to show the silverlight child window at the center of the browser screen

This is not how CP usually works. Most important goal here is to learn and help learning.
You are supposed to try it on your own, and come here when you got stuck with something, with a concrete question about your code, design, etc.
Please have a look to What have you tried?[^] to see a good explanation about what I mean.
Don't forget people here don't get payed. And besides, if we give you a ready-to-go solution, it is not going to help you because you are not going to learn anything from it.

As a generic approach:
Step 1) Do a little research. Google is a good start point
ie:
http://msdn.microsoft.com/en-us/library/system.windows.controls.childwindow.show(v=vs.95).aspx[^]
http://msdn.microsoft.com/en-us/library/system.windows.controls.childwindow(v=vs.95).aspx[^]
http://msdn.microsoft.com/en-us/library/system.windows.uielement.arrange(v=vs.95).aspx[^]
Step 2) Start coding
Step 3) Compile and use the debug to solve little issues
Step 4) When you get a problem you don't know how to solve but at least you tried it, then come back and ask for something concrete with a snippet of the code giving problems

Sorry if this is not the answer you were looking for. But your question is a bit too wide to be answered at the "Quick" Answers. It is better and you get faster help if you make 10 concrete questions about concrete problems, than a big question about a "how-to guide"