Book 1 outlines the fundamental propositions of plane geometry, includ ing the three cases in which triangles are. Euclid then builds new constructions such as the one in this proposition out of previously described constructions. This is the twenty fourth proposition in euclids first book of the elements. In any triangle the sum of any two sides is greater than the remaining one. It is also used frequently in books iii and vi and occasionally in books iv and xi. Although it may appear that the triangles are to be in the same plane, that is not necessary. Indeed, that is the case whenever the center is needed in euclid s books on solid geometry see xi. This is the twentieth proposition in euclids first book of the elements.
He is much more careful in book iii on circles in which the first dozen or so propositions lay foundations. Even in solid geometry, the center of a circle is usually known so that iii. It focuses on how to construct an equilateral triangle. Note that for euclid, the concept of line includes curved lines. Guide now it is clear that the purpose of proposition 2 is to effect the construction in this proposition. So at this point, the only constructions available are those of the three postulates and the construction in proposition i. Euclid does not precede this proposition with propositions investigating how lines meet circles. This is the first proposition in euclids first book of the elements. To construct an equilateral triangle on a given finite straight line. Some of these indicate little more than certain concepts will be discussed, such as def. Guide about the definitions the elements begins with a list of definitions. On a given straight line to construct an equilateral triangle.
Coxeter 10 points out that this proof from pappus is equivalent to invoking the symmetry operations of reflection or rotation of the diagram. Thus, even though nobody was in a position to formalise the concept of. For one thing, the elements ends with constructions of the five regular solids in book xiii, so it is a nice aesthetic touch to begin with the construction of a regular. Use of proposition 23 the construction in this proposition is used in the next one and a couple others in book i.
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