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.

You seemed to have jumped to a conclusion here - are you sure that Catherine does not love Edgar? Chapter 9 is a very key chapter for this discussion, where Catherine discusses her proposal from Edgar with Nelly Dean and talks about how she feels about both Edgar and Heathcliff. It is well worth the pains to examine what she says:

My love for Linton is like the foliage in the woods: time will change it, I'm well aware, as winter changes the trees. My love for heathcliff resembles the eternal rocks beneath: a source of little visible delight, but necessary. Nelly, I am Heathcliff! He's always, always in my mind: not as a pleasure, any more than I am always a pleasure to myself, but as my own being.

We can see by this comparison that Catherine does love Linton, but that she has a very different kind of love for Linton than she does for Heathcliff. It is well worth examining this "love" that Heathcliff and Catherine have. In its description it is made out to be very intense and surreal. Remember that their love is never consummated. It is almost as if their love is parasitic - it causes them to wound each other and to drive each other to extremes.

Of course the depth of their connection is what forces Heathcliff and Linton to a stand-off and results in Catherine's death.