Euclid, Elements I 7©
translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 7: On the same straight-line two different straight-lines respectively equal to the same straight-lines will not be constructed at one and another point on the same sides while having the same limits as the initial lines.

(general diagram)

Diagrams for book 1 proposition 7, following the display, construction, and demonstration(diagram 1) For it is possible, on the same straight-line, AB, let two different straight-lines, AD, DB, respectively equal to the same straight-lines, AG, GB, will not be constructed at one and another point, G and D, on the same sides while having the same limits as the initial lines, so that GA is equal to DA, having the same limit, A, as it and GB to DB, having the same limit, B, as it. (diagram 2) And let GD be joined.
(diagram 3) And so, since AG is equal to AD, an angle, that by AGD, is equal to that by ADG. (diagram 4) Therefore, the angle by ADG is larger than that by DGB. (diagram 5) Therefore, the angle by GDB is much larger than that DGB. (diagram 6) Again, since GB is equal to DB, an angle, that by GDB, is also equal to an angle, that by DGB. But it was also shown much larger, which is impossible. Therefore, on the same straight-line two different straight-lines respectively equal to the same straight-lines will not be constructed at one and another point on the same sides while having the same limits as the initial lines, just what it was required to show.

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