Proof of Euclid's Fifth Postulate

Authors: Jay Yoon

I will present a proof of Euclid’s fifth postulate (I.Post.5) that proves, as an intermediate step, a proposition equivalent to it (I.32); namely, that in any triangle, the sum of the three interior angles of the triangle equals two right angles. The proof that I.32 implies I.Post.5 and vice versa is well-established and will be omitted for the sake of brevity. The proof technique is somewhat unorthodox in that it proves I.33, which states that straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel, before establishing I.32, contrary to the order in which the propositions are demonstrated in Euclid’s Elements. Two triangle congruence theorems, namely the side-angle-side (I.4) and side-side-side congruence theorems (I.8) are employed in order to prove I.33 without recourse to I.Post.5 or any of its equivalent formulations. In addition, a parallelogram is constructed by an unorthodox method; namely, by defining the diagonals upon which the parallelogram’s sides will be determined prior to the sides themselves. The proof assumes the five common notions stated in Book I of The Elements without explicitly making a reference to them when they are used. Furthermore, a figure is presented with color-coded angles and sides, with angles of the same color being equal in measure and sides of both the same color and the same number of tick marks being equal in length. The sides GH and EJ enclosed by brackets are indicated to be equal in length, the reason for the different notation being that the tick marks were used in reference to the halves of GH, namely OG and OH. The tick marks then refer to the parts of GH, and the bracket refers to the whole of GH; the latter is then equated to EJ by I.33, which is proven before its use.

Comments: 5 Pages.

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Submission history

[v1] 2012-05-02 23:20:08

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