#Make multiple congruent segments using math illustrations how to#
But he does not verify any of the ordered field axioms (for instance, he does not explain how to multiply the congruence classes).Įdit. Moise sketches the relation between the two (Hilbert and Birkhoff) sets of axioms and explains, for instance, how to add congruence classes in Hilbert's plane. Moise, "Elementary Geometry from an Advanced Standpoint", Pearson, 1990. Thus, two congruent segments have the same length, again, just by the definition. Two segments $AB, CD$ are congruent in this sense iff there exists an isometry $g$ such that $g(AB)=CD$. a map $g: E \to E$ preserving the distance function. A congruence of this geometry is an isometry, i.e. The length of a segment $AB$ is simply the real number $d(A,B)$. In contrast, if you accept Birkhoff's axiomatization, then the Euclidean plane $E$ comes equipped with a distance function $d: E\times E\to _+$. One needs the two continuity axioms (of Hilbert's axiomatic system) in order to prove that $F$ is isomorphic to the field of real numbers.
Hartshorne, "Geometry, Euclid and Beyond", Springer Verlag, 2000. It turns out that one can (assuming all but the continuity axioms of Hilbert) define an ordered field $F$ whose elements are the congruence classes, you can find a detailed construction in the book: I suspect that this is where your question is coming from. This, of course, is not entirely satisfactory since we are used to the idea that length should be a real number and not an element of some unfamiliar quotient set (coming from some equivalence relation). Given this, the answer is that, by the definition, two segments have the same length iff they are congruent, the congruence class of a segment $AB$ is denoted $$.
If you want to change Hilbert's notion of congruence to something else, you are leaving the realm of Hilbert's axioms and, hence, you would have to list your personal axioms and then ask a new question. (There are alternatives, such as Birkhoff's and Tarski's sets of axioms.) Hilbert's axiomatization has two equivalence relations built in: congruence of segments and congruence of angles. For the purpose of this answer, I assume that you are interested in Hilbertian axiomatization of the Euclidean geometry. So my question is, at first, if my reasoning is correct or if I've misunderstood something of fundamental, and, if I'm right, what are the axioms that we need for exactly represent our intuitive concept of congruence?Ī final notice: this question is related to What really is ''orthogonality''?, where a similar question is posed for angles.įirst of all, in order to make sense of the question one has to specify the axiomatic system via which the (Euclidean) plane is defined. And more, we can have different metric (and different ''length'') for a geometry that satisfies the Hilbert's axioms. Obviously ''to have the same length'' is a congruence relation, but only in a metric space, and, in the same space, different congruence relations can be defined that are not equivalent to ''have the same length''. This construction is essentially the same that Hilbert uses (in the chapter 24) for the construction of an ''Algebra of Segments'', ( where the point $E$ is ''at infinity''). But, clearly, it doesn't satisfies our intuition that congruent segments have the same length. Projecting from $E$ the points $A,B,C$ of $a$ to the other lines, we can define the relation:Ī'B' \equiv A''B'' \quad B'C' \equiv B''C''Īnd we can see that this relation satisfies all axioms of congruence. Look at the second figure, where $a$,$a'$ and $a''$ are straight lines and $E$ is a point that does not belong to these lines. Usually these axioms are illustrated with a figure that use two segments $AB$ and $A'B'$ that have the same length, as in this case.īut I think that this illustration is only dictated by our intuition of congruence as equality of length, and it's not justified by the axioms. I don't report here these axioms that can be found on the book (or on the Wiki page)Īnd essentially say that congruence is an equivalence relation and that congruence is conserved when we add adjacent segments. Two line segments are congruent if they have the same length.īut in Hilbert's Foundations of Geometry congruence is defined without use of metric notions, by the axioms of Group IV (chapter 6). And this really is the definition from Wikipedia: The answer to the question in the title seems an obvious ''Yes by definition !''.
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