Euclid,

ElementsI 23©

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

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Prop. 23: To construct at the given straight-line and the point on it an angle equal to the given rectilinear angle.

(diagram 1) Let the given straight-line be AB and the point on it A, and the given rectilinear angle, that by DGE. It is, in fact, required to construct at the given straight-line, AB, and the point on it, A, a rectilinear angle equal to the given rectilinear angle, that by DGE. (diagram 2) Let there be taken on each of GD, GE points as happens, D, E, (diagram 3) and let DE be joined. (diagram 4) And from three straight-lines, which are equal to three, GD, DE, GE, let a triangle be constructed, AZH, so that GD is equal to AZ and GE to AH and furthermore DE to ZH. (diagram 5) And so, since two, DG, GE, are respectively equal to two, ZA, AH, and a base, DE, is equal to a base, ZH, therefore, an angle, that by DGE, is equal to an angle, that by ZAH. Therefore, at the given straight-line, AB, and the point on it, A, an angle, ZAH, equal to the given rectilinear angle, that by DGE, has been constructed, just what it was required to make.

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