By Rudolf Mehmke
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Additional info for Anwendung der Grassmann'schen Ausdehnungslehre auf die Geometrie der Kreise in der Ebene
2 we find a point E not lying on the line AD (see Fig. 1). Then we apply the axiom A10 to the points C and E. As a result on the line CE we find a point F such that the point E lies in the interior of the segment [CF ]. Let’s draw the lines AE and F B, then consider the triangle F BC. The line AE crosses the line F C at the point E which is an interior point for the segment [F C]. The intersection of the lines AE and BC coincides with the point A which is outside the segment [BC]. Therefore, according to Pasch’s axiom A12, the line AE should cross the side [F B] of the triangle F BC at some interior point G.
But the point Ai+1 does not belong to the segment [Aj , Aj+1 ]. 3). , 1998, 2007. 36 CHAPTER II. AXIOMS OF INCIDENCE AND ORDER. 1. Let A1 , . . , An be a monotonic sequence of points on a straight line and let B be some point of this line coinciding with none of the points A1 , . . , An . Then one can join the point B to the points A1 , . . , An and enumerate the resulting set of points so that the monotonic sequence of points A1 , . . , An+1 will be formed. Proof. Let’s consider the three points A1 , An , and B.
Proof. Assume that pAB and pCD are the mappings of congruent translation on a straight line a and assume that pAB = pCD = p. Let’s define the positive direction on the −→ line a by means of the vector AC and thus define a relation of precedence for the points of this line a. Then A ≺ C. 2: p(A) ≺ p(C), [p(A)p(C)] ∼ = [AC]. But p(A) = pAB (A) = B and p(C) = pCD (C) = D. Hence, −→ −→ A ≺ C and B ≺ D, which means that the vectors AC and BD −→ −→ are codirected. Moreover, [BD] ∼ = [AC], therefore, AC = BD.
Anwendung der Grassmann'schen Ausdehnungslehre auf die Geometrie der Kreise in der Ebene by Rudolf Mehmke