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It's seems as though he is hiding something from you either he is married or has a live in girlfriend or he isn't what he made himself out to be and doesn't want you to find out. You're right, if a man is interested in you he will ask you out and meet you. That is the problem with meeting someone on the internet it's hard to shift out the truth from lies that they think you want to hear. He may be shy but there just seems like there is something he is hiding from you. He may still be married but only on paper and not in heart. What I would do is give him another month and if he still isn't ready to meet I would consider moving on to someone that is willing to meet you and allow you to know the real them and not hide things from you.
It could be that he is shy or afraid about meeting you in person but he shouldn't feel that way for he says you are gorgeous maybe he is afraid of that, maybe he thinks that you won't think he is worth your time. Have you seen pictures of him and are you sure he looks like what he said he looks like? It could be that he is afraid you will meet him and decide that he isn't your type and that means that you have to earn his trust more and give him time stick with giving him another month and see what happens with that. He may just want to get to know you a little bit better because he has his children to think about and doesn't want to just meet a stranger without knowing that your intentions are honorable. When you have children the way you see things and do things change because he has to take into consideration his children.