The Elements

Four geometric diagrams are positioned in the right margin:
1. (Top) Two parallelograms share a common height between two parallel lines. The top line contains points a, b, e, and f; the bottom line contains points c, d, g, and h. Vertical/diagonal lines connect a-c, b-d, e-g, and f-h. This illustrates Proposition 36.
2. (Upper middle) Two triangles, a-b-c and d-b-c, share the same base b-c. They are positioned between parallel lines. The top line contains points g, a, k, d, f, and h. Parallelograms are constructed around the triangles using these points to show that the triangles are half the area of the parallelograms. This illustrates Proposition 37.
3. (Lower middle) Two triangles, a-b-c and d-e-f, are drawn on equal bases b-c and e-f along the same bottom line. They are positioned between parallel lines, with the top line containing points a, k, d, and h. Parallelograms are constructed on each base. This illustrates Proposition 38.
4. (Bottom) A diagram illustrating two triangles with the same base b-c and equal areas. One triangle is a-b-c, and the other is d-b-c. A line a-e-f is drawn through vertex a to show that if the triangles are equal, vertex d must also lie on the same parallel line. This illustrates Proposition 39.

Therefore, having removed the common line f-g, a-g will remain equal to f-b, because by the preceding original: "p̄ pcedentē" refers to the previous step in the proof again a-c is equal to f-e, and angle b-f-e is equal to angle g-a-e by the second part of Proposition 29—namely, the exterior angle is equal to the interior angle. Thus, by Proposition 4, triangle a-c-g will be equal to triangle f-e-b. Therefore, if the irregular quadrilateral figure g-c-f-e is added to both, the surface The word "superficies" here refers to the area of the figure a-c-f-e will be equal to the surface g-c-b-e, which is what was to be proved.

Now, let the line c-g cut the line a-b at point f, as appears in the second figure. By a similar argument as before, the two triangles a-c-f and f-c-b will be equal, wherefore by adding triangle f-c-e to both, the proposition is clear.

In a third way, let line c-g cut line a-b between the two points f and b, as appears in the third figure; and let it cut line f-e such as at point k. Because by a similar argument as before line a-f is equal to line g-b, and by making line g-f common, a-g will be equal to f-b, and triangle a-g-e equal to triangle f-c-b. Therefore, by adding triangle c-k-e to both and subtracting triangle f-k-g from both, the surface a-c-f-e will be equal to the surface g-c-b-e, which is what was to be proved.

Proposition .36.

It is necessary that all parallelograms established on equal bases and within the same parallel lines are equal.
¶ A parallelogram Parallellogramū: a four-sided figure where opposite sides are parallel is defined as a surface of equidistant sides. Let there be two surfaces of equidistant sides, a.b.c.d. and e.f.g.b., established between two equidistant lines, which are a.f. and c.b., and upon equal bases, which are c.d. and g.b. I say that they are equal. For if I draw two lines c.e. and d.f., the surface c.d.e.f. will be of equidistant sides by Proposition 33, because e.f. is equal and equidistant to c.d., for each of them is equal to g.b. Since, therefore, by the premise, each of the two surfaces a.b.c.d. and e.f.g.b. is equal to the surface c.d.e.f., they shall be equal to each other, which is what was to be proved.

Proposition .37.

All triangles that are established upon the same base and between two parallel lines are equal to one another.
¶ Let there be two triangles, a.b.c. and d.b.c., established upon the base b.c. between two lines a.e. and b.f. which are equidistant parallel. I say they are equal. For I shall draw c.g. parallel to a.b. and c.h. parallel to d.b. by Proposition 31. There will then be two surfaces a.b.c.g. and d.b.c.h., which are equal by Proposition 35. And because the said triangles are half of them, by the corollary of Proposition 34, they will be equal to each other by the common science original: "cōez sciam," referring to a Common Notion or Axiom in Euclid which states that those things whose wholes are equal are also themselves equal in their halves; thus the proposition is clear.

Proposition .38.

If two triangles fall upon equal bases and between two parallel lines, it is necessary that they be equal.
¶ Let there be two triangles, a.b.c. and d.e.f., established upon equal bases b.c. and e.f. and between equidistant lines a.g. and b.h. I say they are equal. For I shall draw c.k. parallel to a.b. and f.l. parallel to e.d. There will then be two surfaces a.b.c.k. and d.e.f.l., equal by Proposition 36. And since the said triangles are half of them by the corollary of Proposition 34, they will be equal by the aforementioned common science.

Proposition .39.

All pairs of equal triangles, if they fall upon the same base and on the same side, will be between two parallel lines.
¶ Let there be two triangles, a.b.c. and d.b.c., established upon the base b.c. on one and the same side; and let them be equal. I say they are between equidistant lines. This is the converse of Proposition 37. From point a, I shall draw a line parallel to line b-c. If this line passes through point d, the proposition is evident. But if it passes above or below: let it first pass above, and let it be a.e., and I shall extend b.d. until it cuts...