Why did Rex actually believe that Charlie was his father?

Rex didn't get a private investigator license for nothing. And how did Roxy explain the coincidence involving Charlie's surname? That Charlie is what Dr. Walter Balsom's brother or something? Last time I checked, I thought Rex pretended to buy it so that he could catch Charlie in the act when he finally slips up based on the eventual inconsistencies in the lies that he told about how he and when he met Roxy etc.