On Saturday, February 15, 2014 6:04:10 AM UTC+2, Ross A. Finlayson wrote: > On 2/14/2014 6:00 PM, 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 > > > > > > > > > > > > > (proximal, proximity) > > > > The circle fills space uniformly with the least perimeter for > > area so they are the same. It is half-way then for the > > equilateral, because the angles are equi-angular, in the > > evolution of angles, they are equal. > > > > Sorry, this is a rather poor geometric construction, I'd be > > most surprised to hear of something like that in Euclid. > > > > Then, it would because the radii at a distance from the line > > with the parallel line, with the adjustable compass from > > drawing the parallel lines, would form the triangles with that > > height that form the equilateral triangle (that the side is > > equal, to find then the parallel line at the height of the > > equilateral triangle. This is finding the center from > > shifting that up then finding the mid-point of that, bisecting > > the segment with the compass and edge, then halving and > > doubling, for the projection of the triangle that is of the > > parallel lines, to the orthogonal parallel distance that is of > > the third side of the equilateral triangle, through that. > > > > Or, having three compasses.
What absolute rubbish. I can tell you don't understand the Elements at all!