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16.4 If F is a Pythagorean ordered field, prove the triangle inequality in the correspond- ing plane II, namely, if A, B, C are three points in II, then dist(A,C) ≤ dist(A, B) + dist(B, C), and equality holds if and only if A, B, C are collinear and B is between A and C.

16.4 If F is a Pythagorean ordered field, prove the triangle inequality in the correspond- ing plane II, namely, if A, B, C are three points in II, then dist(A,C) ≤ dist(A, B) + dist(B, C), and equality holds if and only if A, B, C are collinear and B is between A and C.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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16.4 If F is a Pythagorean ordered field, prove the triangle inequality in the correspond- ing plane II, namely, if A, B, C are three points in II, then dist(A, C) ≤ dist(A, B) + dist(B, C), and equality holds if and only if A, B, C are collinear and B is between A and C.
Transcribed Image Text:16.4 If F is a Pythagorean ordered field, prove the triangle inequality in the correspond- ing plane II, namely, if A, B, C are three points in II, then dist(A, C) ≤ dist(A, B) + dist(B, C), and equality holds if and only if A, B, C are collinear and B is between A and C.
However, our field F may not have square roots in it, so we will use instead the "distance-squared" function dist²(A, B) = (a₁ − b₁)² + (a₂ − b₂) ². This will give the same notion of congruence. Definition Two line segments AB and CD in the Cartesian plane over an ordered field F are congruent if dist²2 (A, B) = dist² (C, D). Since congruence is defined using the function dist2 from line segments to the field, the axiom (C2), transitivity of congruence, will be obvious. Notice that because of the ordering on F, if A, B are distinct points, then dist²(A,B) > 0.
Transcribed Image Text:However, our field F may not have square roots in it, so we will use instead the "distance-squared" function dist²(A, B) = (a₁ − b₁)² + (a₂ − b₂) ². This will give the same notion of congruence. Definition Two line segments AB and CD in the Cartesian plane over an ordered field F are congruent if dist²2 (A, B) = dist² (C, D). Since congruence is defined using the function dist2 from line segments to the field, the axiom (C2), transitivity of congruence, will be obvious. Notice that because of the ordering on F, if A, B are distinct points, then dist²(A,B) > 0.
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