Euclid,

ElementsI 22©

translated by Henry Mendell (Cal. State U., L.A.)

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Prop. 22: To construct a triangle from three straight-lines which are equal to three, the given straight-lines. But it is required that two, taken in any way, be larger than the remaining[, since the two sides of any triangle, taken in every way, are also larger than the remaining. (i 20)

(diagram 1) Let the given three straight-lines be A, B, G, (diagram 2) for which let the two, taken in any way be larger than the remaining, A, B than G and A, G than B and furthermore B, G than A. It is, in fact, required to construct a triangle from equals to A, B, G. (diagram 3) Let some line be displayed that’s finite along D but infinite along E, (diagram 4) and let an equal, DZ, to A, be positioned and an equal, ZH, to B, and an equal, HQ, to G. (i 3) (diagram 5) And with a center, Z, and a distance, ZD, let a circle be described, DKL. (diagram 6) Again, with a center, H, and a distance, HQ, let a circle be described, KLQ, and let KZ, KH be joined. (diagram 7) I say that from three straight-lines, those equal to A, B, G, a triangle has been constructed, KZH.

For since point Z is center of circle DKL, ZD is equal to ZK. But ZD is equal to A. Therefore, KZ is also equal to A. Again, since point H is center of circle LKQ, HQ is equal to HK. But HQ is equal to G. Therefore, KH is also equal to G. But ZH is also equal to B. Therefore, the three straight-lines, KZ, ZH, HK, are equal to three, A, B, G. Therefore, a triangle, KZH, has been constructed from three straight-lines, KZ, ZH, HK, which are equal to three, the given straight-lines, A, B, G, just what it was required to make.

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