I think that what might be going on is that your wife feels that what you are doing is making her feel insecure, like she's somehow not part of the process and that it might be some kind of criticism of her somehow. You've got to get that straightened out. What you do for yourself is just as important as what you do for the family. Instead of feeling upset they need to feel inspired. They are a part of your improvement 100% emotionally, if not physically. And if she feels left out try to include her in your activities.
Maybe she's not going to attempt a triathlon or something but ride with her regularly. Get her a bike she truly likes, one of her choosing.
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