Euclid,

ElementsI 45©

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

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Prop. 45: To construct a parallelogram equal to the given rectilinear-figure in the given rectilinear angle.

(diagram 1) Let the given rectilinear-figure be ABGD, the given angle E. It is, in fact, required to construct a parallelogram equal to rectilinear-figure ABGD in the given angle E.

(diagram 2) Let DB be joined, (diagram 3) and let a parallelogram, ZQ, be constructed equal to triangle ABD in the angle by QKZ, which is equal to E. (diagram 4) Let a parallelogram, HM, equal to triangle DBG in angle HQM, which is equal to E, be applied along straight-line HQ. And since angle E is equal to each of those by QKZ, HQM, therefore, the angle by QKZ is also equal to that by HQM. (diagram 5) Let a common, the angle by KQH, be added. Therefore, the angles by ZKQ, KQH are equal to those by KQH, HQM. But the angles by ZKQ, KQH are equal to two right-angles. Therefore, the angles by KQH, HQM are also equal to two right-angles. In fact, at some straight-line, HQ, and the point on it, Q, two straight-lines, KQ, QM, not positioned on the same sides, make successive angles equal to two right-angles. Therefore, KQ is on a straight-line with QM. (diagram 6) And since a straight-line, QH, falls into parallels, KM, ZH, the alternate angles, those by MQH, QHZ, are equal to one another. . (diagram 7) Let a common, the angle by QHL, be added. Therefore, the angles by MQH, QHL are equal to those by QHZ, QHL. But those by MQH, QHL are equal to two right-angles. Therefore, ZH is on a straight-line with HL. (diagram 8) And since ZK is equal and parallel to QH, but also QH to ML, therefore, KZ is also equal and parallel to ML. And straight lines, KM, ZL, join them. Therefore, KM, ZL, are also equal and parallel. (diagram 9) Therefore, KZLM is a parallelogram. And since triangle ABD is equal to parallelogram ZQ, and DBG to HM, therefore, a whole, rectilinear-figure ABGD, is equal to a whole parallelogram KZLM. Therefore, a parallelogram, KZLM, equal to the given rectilinear-figure, ABGD, in an angle, that by ZKM, which is equal to the given, E, has been constructed, just what it was required to make.

The construction presupposes that any polygon ABGD can be triangulated, divided up into triangles, for which cf. also vi 18 and subsequently vi 20, where Euclid goes through a construction of a sort for pentagons. Although it may be obvious that one can always do this, it is not so trivial for non-convex polygons, while the construction here and at vi 18, involving a quadrilateral and one construction, certainly of a convex figure in the diagram, and that at vi 20 of a convex pentagon lack an expected generality. The ability to always break up a figure (finite for rectilinear figures and infinite for curvilinear figures) into triangles is an imporant background assumption of 4th century BCE mathematics and is reflected in the *Elements*, but as strikingly in Aristotle's discusion of the definition of the soul (*De anima* II 3).

As is often remarked, this theorem, with its two companions in vi, is a first step in a rich ancient discussions of the study of areas and applications of areas, taken up in book ii. top