The Elements

Three geometric diagrams are positioned in the right margin to illustrate the proofs:
1. (Top) A construction for Proposition 42 showing a rectangle and a triangle of equal height. Points are labeled b, g, f on the top line; c, e on a middle horizontal line; a to the left; and d at the bottom right. A diagonal line connects b to d.
2. (Middle) A diagram for Proposition 43 showing a large rectangle a-b-d-c divided into four smaller parallelograms by a horizontal line e-f and a vertical line g-h intersecting on a diagonal b-c.
3. (Bottom) A faint geometric construction for Proposition 44, showing several intersecting lines and a triangle used for the application of areas to a specific line.

...a triangle having been constructed, to which a parallelogram will be double by the preceding [proposition], and it itself is equal to the other triangle by Proposition 38.

¶ Proposition 42

To construct a parallelogram literally "a surface of equidistant sides" whose angle is equal to a given angle, and whose area is equal to a given triangle.
¶ Let the given angle be a and the given triangle be b-c-d. I wish to describe a parallelogram equal to triangle b-c-d, in which each of the two angles positioned opposite each other is equal to angle a. I divide the base c-d in half at point e, and I draw line b-e. From point b, I draw b-f parallel to c-d. By Proposition 38, triangle b-c-e is equal to triangle b-e-d, and therefore triangle b-e-d is half of the total triangle b-c-d. Then, at point e on line d-c, I construct angle d-e-g equal to angle a, and I complete the parallelogram g-e-d-f. Because this figure is double the triangle b-e-d by the preceding proposition, it will also be equal to triangle b-c-d by this common notion original: "cōam scienciam," referring to Euclid's Axioms: those things whose halves are equal are also equal to each other. For triangle b-e-d is half of both; therefore, we have described the parallelogram g-e-d-f equal to triangle b-c-d, in which each of the two opposite angles g-e-d and d-f-g is equal to angle a, which was proposed.

¶ Proposition 43

In every parallelogram, the complements term: "supplementa," the areas remaining in a parallelogram when smaller parallelograms are constructed along the diagonal of those parallelograms which are about the diagonal must be equal to one another.
¶ Let there be a parallelogram a-b-c-d in which I draw the diagonal b-c. I draw e-f parallel to both sides a-b and c-d, intersecting the diagonal at point k. From there, I draw k-g parallel to both sides a-c and b-d, extending it until it intersects both sides a-b and c-d; let the whole line be g-k-h. The whole parallelogram a-b-c-d is thus divided into four parallelograms, two of which—namely e-c-k-h and g-k-b-f—are said to stand about the diagonal c-b because the diagonal passes through their centers. Therefore they are "about the diagonal." The remaining two—namely a-e-g-k and k-h-f-d—are called the complements. It is said that these two complements are equal. For the two triangles a-b-c and c-d-b are equal by the conclusion of Proposition 34. Similarly, the two triangles g-k-b and f-k-b are equal by the same conclusion. Likewise, the two triangles c-e-k and k-h-c are equal by that same conclusion. Therefore, if the two triangles b-g-k and k-e-c are subtracted from the whole triangle a-b-c, and the two remaining triangles b-f-k and k-c-h are subtracted from the other whole triangle c-d-b, the remainders will be equal by common notion. These remainders are the two aforementioned complements; which is what was proposed.

¶ Proposition 44

Given a straight line, to construct upon it a parallelogram whose angle is equal to a given angle, and whose area is equal to a given triangle.
¶ To construct a parallelogram upon a given line means to make that line one side of the figure. Let the given line be a-b, the given angle c, and the given triangle d-e-f. I wish to construct upon line a-b a parallelogram so that a-b is one of its sides, each of its opposite angles is equal to angle c, and the total area is equal to triangle d-e-f. This differs from Proposition 42 because here one side of the figure to be described is given (line a-b), whereas there no side was specified. Therefore, when I wish to do this, I join line a-g...