There's nothing the matter with you. It's normal to be upset that he replaced you so soon after breaking up with you! It makes you feel like you were nothing to him!

I don't blame you.
BUT -- there are other ways to deal with your anger besides writing both of them. Don't do that. Instead, write in a journal about your feelings, or write poems. But don't contact either one of them. All that does is stir up trouble, and perhaps get YOU in trouble for harassment!
Chances are his "new love" won't last long anyway. But I would NOT take him back, because he's just NOT STABLE. Since he's recently been diagnosed with BiPolar disorder, he's already had 2 girlfriends -- you and this other chick. Does that sound stable? NO. So until he gets both feet on the ground, stay away from him.
God bless and please take care. Hugs, Lee