We look back to concept and dependency graphs of Euclid’s Elements, Book 1, that show the deductive relationships among its propositions. We claim that Book 1 does not have one overall coherent structure but rather is organized around two propositions—1.45 and 1.47. In other words, Book 1 has a dual core. For the latter proposition, the Pythagorean Theorem, we constructed a rooted in-tree graph to help visualize its role and proof.