You have asked an interesting question. I guess we don't actually know whether Edgar Linton does know this or not - he is certainly not privy to the conversation between Catherine and Nelly Dean which Heathcliff overhears where Catherine clearly states her relationship with Heathcliff in mystical tones: "I am Heathcliff." Obviously Edgar would have been aware of the special place that Heathcliff had in Cathy's affections, but at the same time with his departure we can perhaps infer that he thought he had triumphed in the competition for Cathy's affections.
What we can definitely see however is that when Heathcliff returns Edgar almost from the start realises that their former attachment to each other has never been broken off and, unable to force Cathy to decide between him and Heathcliff, retreats to his study. Interestingly, Linton at first glance represents everything Heathcliff is not: he is refined, educated, cultured etc. Yet at the same time when we look at his character we see that his inability to force Catherine into making a decision, his treatment of his sister after she eloped with Heathcliff and his failure to protect his daughter suggests a certain moral weakness in his character.