2nd Stage, Lecture 6: Euclidean Geometry

Euclidean Geometry
Axioms
Ax1: it is possible to draw the straight line from any point to any another point.
Ax2: It can expand line from both sides into any length.
Ax3: It can draw circle if one knows its center and it radius.
Ax4: All 90o angles are equal.
Ax5: If two lines are intersected by a third line and the sum of the inner angles that are lie on the same side of the intersector line less than 90o then the lines can be drown into any length and they meet in the same side of the intersected that the sum of the angles less than 90o.

Theorem 1. It can draw triangle has equaled sides on the known straight line and limited length.

Proof. Let xy is the straight line. From Ax3, there is a circle, its center x and its radius xy. Also, from Ax3 it can draw another circle, its center y and its radius xy. The intersection point of two circles is A.
From Ax1, there are the lines xA and yA. From definition of a circle, xA=xy and yA=xy.
So, xy=xA=yA. Thus, the triangle has equaled sides.