 The proof of prop. 1 has famously been objected for having a gap. Given a segment \(AB\), an equilateral triangle is constructed by drawing two circles with \(AB\) as radius, one centered on \(A\) and the other centered on \(B\). The objection is that Euclid has no continuity principle, hence the existence of point \(C\) (the intersection of both circles) is not warranted. But such an objection ignores the proper role of diagrammatic reasoning in Euclid.  Manders (2008) has provided an insightful analysis of how information is obtained from text and diagram in the first books of the Elements, in careful distribution of roles, which led him to distinguish between exact information (from text only) and co-exact information (taken from the diagram). Notice that a diagram is not just a figure, but a reasoning tool, a semiotic element with rules of manipulation – and, according to Manders, careful (implicit) rules for the extraction of information. For further details on this topic, see the entry on  diagrams, sections 4.2 and 4.3. 