On Saturday, February 15, 2014 4:00:41 AM UTC+2, Ken Quirici wrote: > According to Wikipedia, Book 1 Proposition 1 (constructing equilateral triangle on given line segment) > > is missing a premise - that if two different circles share a radius (that is, a line segment connecting > > their centers is a radius for both circles). > Could somebody provide a proof of this premise? > Regards, > > Ken
The Greek implies that the circles so drawn are equal, for otherwise the construction such that the sides of the equilateral triangle are equal would not be possible. It follows logically from the context. There is no missing premise. In the construction you must use circles with equal radii for otherwise you cannot construct an equalitareal triangle. This proposition is used to construct angles such as 60 degrees and 30 degrees.