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Saccheri-Legendre Theorem

    This topic defines a Dedekind cut and proves the Dedekind Axiom implies the Archimedian Axiom. After introducig the measure of a segment and an angle, the triangular inequality and the Saccheri-Legendre Theorem are proven.

Proposition (Saccheri-Legendre) The sum of the degree measures of the three angles of a triangle is less than or equal to saccheri legendre theorem _gr_1.gif]

    Proof. Assume, on the contrary, that the angle sum of saccheri legendre theorem _gr_2.gif] is greater than saccheri legendre theorem _gr_3.gif] say saccheri legendre theorem _gr_4.gif] where saccheri legendre theorem _gr_5.gif] is a positive number. By the Equilvalent Angle Sum Proposition, replace saccheri legendre theorem _gr_6.gif] with another traingle that has the same angle sum as saccheri legendre theorem _gr_7.gif] but in which one of the angles has at most half the number of degrees as saccheri legendre theorem _gr_8.gif] Repeat the procedure to get another triangle that has the same angle sum as saccheri legendre theorem _gr_9.gif]  and has an angle that is one-quarter the degree measure of   saccheri legendre theorem _gr_10.gif] The Archimedian Principle for real numbers guarantees that if this process is repeated enough times, eventually a triangle that has angle sum saccheri legendre theorem _gr_11.gif] and with one angle with degree measure at most saccheri legendre theorem _gr_12.gif] Thus, the sum of the degrees measures of the other two angles will be greater than or equal to saccheri legendre theorem _gr_13.gif] contradicting the Two Angles In A Triangle Proposition. saccheri legendre theorem _gr_14.gif]

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Saccheri Legendre Theorem
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/saccheri-legendre-theorem.html
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