How can I mention a personal tragedy in a statement for PhD?
Yes, you should absolutely mention it. If you do not, the program will not have the necessary context to evaluate your application fairly -- which is bad for them and worse for you.

You should also mention that the program already agreed to consider your application without the GRE -- as you say, it's anyone's guess whether they will remember this concession if you don't remind them. In fact, I worry that a well-meaning secretary might trash your application in any case (as it appears incomplete), so it might be worth indicating this in a more visible place or even calling to confirm that your application was passed to the committee. 

Now, for how to mention it: concisely. 1-2 sentences is all you need. "I performed poorly during this brief period because my brother died" is all they need to know -- you don't need to dress that statement up too much or say much more than that. 


If you go to town making this statement too long, it will sound like an excuse. So, be very clear, direct, and above all concise.
The one thing that might be worth adding is a reason to be confident that you will not continue to have grief-related problems as a PhD student (i.e., give some evidence that things are improving). 


I do not recommend putting this in the last paragraph as you suggest. The last paragraph should send them off on a high note ("declare victory" and look to the future). Instead, put this into the relevant body paragraph. 

Sorry to hear about your brother.