Euclid, Elements of Geometry, Book I, Proposition 34
(Edited by Dionysius Lardner, 1855)

Proposition XXXIV. Theorem.
[Euclid, ed. Lardner, 1855, on Google Books]

(151) The opposite sides (A B and C D, A C and B D) of a parallelogram (A D) are equal to one another, as are also the opposite angles (A and D, C and B), and the parallelogram itself is bisected by its diagonal (A D).

For in the triangles C D A, B A D, the alternate angles C D A and B A D, C A D and B D A are equal to one another (XXIX), and the side A D between the equal angles is common to both triangles; A B C D therefore the sides C D and C A are equal to A B and B D (XXVI), and the triangle C D A is equal to the triangle B A D, and the angles A C D and A B D are also equal; and since the angle A C D with C A B is equal to two right angles (XXIX), and A B D with C D B is equal to two right angles, take the equals A C D and A B D from both, and the remainders C A B and C D B are equal.

(152)   Cor. 1.—If two parallelograms have an angle in the one equal to an angle in the other, all the angles must be equal each to each. For the opposite angles are equal by this proposition, and the adjacent angles are equal, being their supplements.

(153)   Cor. 2.—If one angle of a parallelogram be right, all its angles are right; for the opposite angle is right by (151), and the adjacent angles are right, being the supplements of a right angle.

(154)   Both diagonals A D, B C being drawn, it may, with a few exceptions, be proved that a quadrilateral figure which has any two of the following properties will also have the others:

1° The parallelism of A B and C D.

2° The parallelism of A C and B D.

3° The equality of A B and C D.

4° The equality of A C and B D.

5° The equality of the angles A and D.

6° The equality of the angles B and C.

7° The bisection of A D by B C.

8° The bisection of B C by A D.

9° The bisection of the area by A D.

10° The bisection of the area by B C.

These ten data combined in pairs will give 45 distinct pairs; with each of these pairs it may be required to establish any of the eight other properties, and thus 360 questions respecting such quadrilaterals may be raised. These questions will furnish the student with a useful geometrical exercise. Some of the most remarkable cases are among the following corollaries:

The 9th and 10th data require the aid of subsequent propositions.

(155)   Cor. 3.—The diagonals of a parallelogram bisect each other. A B C D E

For since the sides A C and B D are equal, and also the angles C A E and B D E, as well as A C E and D B E, the sides (XXVI) C E and B E, and also A E and E D are equal.

(156)   Cor. 4.—If the diagonals of a quadrilateral bisect each other, it will be a parallelogram.

For since A E and E C are respectively equal to D E and E B, and the angles A E C and D E B (XV) are also equal, the angles A C E and D B E are equal (IV); and, therefore, the lines A C and B D are parallel, and, in like manner, it may be proved that A B and C D are parallel.

(157)   Cor. 5.—In a right angled parallelogram the diagonals are equal. A B C D

For the adjacent angles A and B are equal, and the opposite sides A C and B D are equal, and the side A B is common to the two triangles C A B and A B D, and therefore (IV) the diagonals A D and C B are equal.

If the diagonals of a parallelogram be equal, it will be right angled.

For in that case the three sides of the triangle C A B are respectively equal to those of D B A, and therefore (VIII) the angles A and B are equal. But they are supplemental, and therefore each is a right angle.

(158)   The converses of the different parts of the 34th proposition are true, and may be established thus:

If the opposise sides of a quadrilateral be equal it is a parallelogram. A B C D

For draw A D. The sides of the triangles A C D and A B D are respectively equal, and therefore (VIII) the angles C A D and A D B are equal, and also the angles C D A and D A B. Hence the sides A C and B D, and also the sides A B and C D are parallel.

Hence the lozenge is a parallelogram, and a square has all its angles right.

If the opposite angles of a quadrilateral be equal, it will be a parallelogram.

For all the angles together are equal to four right angles (134); and since the opposite angles are equal, the adjacent angles are equal to half the sum of all the angles, that is, to two right angles, and therefore (XXVIII) the opposite sides are parallel.

If each of the diagonals bisect the quadrilateral, it will be a parallelogram. A B C D E

This principle requires the aid of the 39th proposition to establish it. The triangles C A D C B D are equal, each being half of the whole area, therefore (XXXIX) the lines A B and C D are parallel. In the same manner D A B and D C B are equal, and therefore A C and B D are parallel.

(159)   The diagonals of a lozenge bisect its angles.

For each diagonal divides the lozenge into two isosceles triangles whose sides and angles are respectively equal.

(160)   If the diagonals of a quadrilateral bisect its angles, it will be a lozenge.

For each diagonal in that case divides the figure into two triangles, having a common base placed between equal angles, and therefore (VI) the conterminous sides of the figure are equal.

(161)   To divide a finite right line A L into any given number of equal parts. B C D E A b c d L X m

From the extremity A draw any right line A X of indefinite length, and take upon it any part A B. Assume B C, C D, D E, &c. successively equal to A B (III), and continue in this manner until a number of parts be assumed on A X equal in number to the parts into which it is required to divide A L. Join the extremity of the last part E with the extremity L, and through B C D &c. draw parallels to E L. These parallels will divide A L into the required number of equal parts.

It is evident that the number of parts is the required number.

But these parts are also equal. For through b draw b m parallel to A E, and b c is a parallelogram; therefore b m is equal to B C or to A B. Also the angle A is equal to the angle c b m, and b B to b c m. Hence (XXVI) b and b c are equal. In like manner it may be proved, that b c and c d are equal, and so on.

(162)   Parallelograms whose sides and angles are equal are themselves equal. For the triangles into which they are divided by their diagonals have two sides and the included angles respectively equal, and are therefore (IV) equal, and therefore their doubles, the parallelograms, are equal.

(163)   Hence the squares of equal lines are equal.

(164)   Also equal squares have equal sides. For the diagonals being drawn, the right angled isosceles triangles into which they divide the squares are equal; the sides of those triangles must be equal, for if not let parts be cut off from the greater equal to the less, and their extremities being joined, an isosceles right angled triangle will be found equal to the isosceles right angled triangle whose base is the diagonal of the other square (IV), and therefore equal to half of the other square, and also equal to half of the square a part of which it is; thus a part of the half square is equal to the half square itself, which is absurd.

Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)

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