By David Gans
Booklet through Gans, David
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Additional resources for An Introduction to Non-Euclidean Geometry
Actually, it is triangle ABC plus its interior that is subdivided into the two triangles plus their interiors. 60 III. PARALLELS WITH A COMMON PERPENDICULAR If d, dl9 d2 are the defects of triangles ABC, ADC, BDC, then, as we now show, d=d1+d2. y + z) - (s + x) = 1 8 0 ° - ( r + / + j + z) =
To give one more example, let us note that by negating Substitute (h) in Chapter I, Section 6, which states that similar triangles exist which are not congruent, we obtain the statement: Similar triangles are always congruent. This is a fact of hyperbolic geometry. This procedure of negating substitutes for Postulate 5 is obviously useful in quickly providing us with salient facts of hyperbolic geometry. Sometimes the facts thus obtained are complete and tell us everything we could wish to know about the corresponding situations.
33). This proves the second part of the theorem. On noting that these quadrilaterals can also be viewed so that AE, DF and EB, FC are arms, we again use Theorem 33, obtaining and DF>AE FC > EB. Combining these inequalities gives DF+FC>AE + EB, or DC > AB, which completes the proof of the theorem. Lines AB, CD in the preceding discussion are parallel since they have a common perpendicular, and AD, EF are two distances from line CD to line AB that are unequal. Thus we have proved by a direct geometrical procedure that parallel lines exist which are not equidistant from one another.
An Introduction to Non-Euclidean Geometry by David Gans