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.

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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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