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(APMO 1990) Let $G$ is a graph that satisfies the following conditions. (1) No point has edges to all other points, (2) there are no triangles, (3) given any two points $A,B$ s.t. there is no edge AB, there is exactly one point $C$ that there are edges AC and BC. Prove every point has the same degree.
I found a proof online, but it uses common complement, which doesn't make sense to me. I've tried considering the $N_a,N_b$ of two points A, B, where there isn't an edge $AB$ and proving them having no edges between $N_a,N_b$ without, and their common point $C$, but then I'm stuck.