This is a "normal" thing, there was a HN article about it some weeks back: often if someone has their mind made up about something, often no amount of facts, figures, logic and proof will change their minds. They may even see these things as personal attacks. Facts are, sadly, not an effective way to convince people.
I'm currently reading the book "Never split the difference" by Chris Voss, a book on negotiation by a former FBI hostage negotiator. In it, he talks at length about why traditional negotiation techniques don't work: they assume that both sides are rational and can be persuaded through facts and value propositions. People aren't rational, they're emotional and to get through to someone, you need to get through to their emotional mind. The book then talks about different techniques he's used to do this, but the fundamental foundation seems to be to listen to what the other person is saying -- really, really listen -- and put yourself into their emotional shoes so you can understand not just what emotion they're feeling, but also why. He then uses rather simple techniques: he verbalises the emotions he thinks their feeling and the reasons. This forces the other person to acknowledge them (or correct him if he got it wrong), which has a calming effect, provides you with more information to work with and often makes the person realise if they're being unreasonable. That's only one technique used, of course, but you can see how he's not trying to engage their logical factual minds, but rather find a way to put out the emotional fires going on in the persons mind.
I think these same principles apply when trying to convince someone of anything. If they have an emotional stake in the debate, then facts and logic aren't going to cut it, but talking to their emotions hostage-negotiator-style just might.