Let BC be an arbitrary line segment connecting points B and C in the Euclidian plane. Let E be an arbitrary point in the Euclidian Plane.
One can construct an equilateral triangle DBC with from a given line BC. (prop. 1)
One can construct a circle with center C and radius CE. (Postulate 3)
One can extend line DC to be longer then CE. (Postulate 2)
The Circle Centered at C with Radius CE intersects line DC at, at most 2 points H, J. (Postulate 3 or Definition of Circle)
WLOG name H and J s.t. DH is shorter or the same in length to the line DJ. (Postulate 1)
(This now becomes the alternate case presented in class)
One can construct a circle with center D and radius DH.
One can extend Line DB to be longer then DM (Postulate 2)
The Circle Centered at D with Radius DH intersects line DB at most two points M, N. (Postulate 3 and Definition of circle)
WLOG name M s.t. the length of line HM is less then or the same as HN. (Postulate 1)
Line CE is the same length as Line CH as they are both radiuses of the same circle (Definition of circle)
Line DH is the same length as line DM as both lines are radiuses of the same Circle (Definition of Circle)
Line DC is the same length as Line DB (Definition of equilateral triangle)
Line CH is the same length as line BM (Common notion 1)
Line CE is the same length as BM (Common notion 1)
"Space" means: Interior of some Euclidean Sphere (call it the “fundamental sphere”)
"Point" means: point in the interior of the fundamental sphere.
"Line" means: the portion of a circle passing though the interior of the fundamental sphere and orthogonal to the boundary.
"Circle" means: The set of coplanar points equidistant from a given fixed point, the center.
The idea of a fundamental circle is one of convention. If you choose to you Euclidean geometry then the fundamental circle is one of the basic curves of the system. It is the set of all points equidistant from a point. One...