# neutral1 - Every line segment has exactly one midpoint...

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Math402 Neutral Geometry Here is a summary of the neutral geometry, i.e. geometry based on Euclid/Hilbert/Birkhoﬀ/School axiomatic system without the Parallels Axiom ( i.e. without Euclid V or Playfair or Wallis Axiom). Euclidean and hyperbolic geometry are special cases of neutral geometry. Axioms. Either of Euclid/Hilbert/Birkhoﬀ/School axiomatic systems without the Parallels Axiom. The particular choice does not matter after basic theorems are proven. We use school system: Hilbert + length and angle measure - Playfair. Basic Theorems. Here are some basic (axiom-type) theorems. Such statements could be trivial in one axiomatic system, but non-trivial in another. Theorem.
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Unformatted text preview: Every line segment has exactly one midpoint. Theorem. Every angle has exactly one bisector. Theorem. Vertical angles are congruent. Theorems. SAS, ASA, SSS triangle congruences. Theorems. Isosceles Triangles Theorem. Two angles of a triangle are congruent if and only if the opposite sides are congruent. Theorem. In a triangle larger side is opposite larger angle and larger angle is opposite larger side. Triangle Inequality Theorem. The sum of lengths of two sides of a triangle is greater than the length of the third side. Exterior Angle Theorem. An exterior angle of a triangle is greater than either of the two nonadjacent interior angles....
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