Euclid, Elements of Geometry, Book I, Proposition 43
(Edited by Dionysius Lardner, 1855)

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

(195) In a parallelogram (A C) the complements (A K and K C) of the parallelograms about the diagonal (E G and H F) are equal.

Draw the diagonal B D, and through any point in it K draw the right lines F E and G H parallel to B C and B A; A B C D E F G H K then E G and H F are the parallelograms about the diagonal, and A K and K C their complements.

Because the triangles B A D and B C D are equal (XXXIV), and the triangles B G K, K F D are equal to B E K, K H D (XXXIV); take away the equals B G K and K E B, D F K and K H D from the equals B C D and B A D, and the remainders, namely, the complements A K and K C, are equal.

(196)   Each parallelogram about the diagonal of a lozenge is itself a lozenge equiangular with the whole. For since A B and A D are equal, A B D and A D B are equal (V). But E K B and A D B are equal (XXIX), therefore E K B and E B K are equal, therefore E K and E B are equal, and therefore E G is a lozenge. It is evidently equiangular with the whole.

(197)   It is evident that the parallelograms about the diagonal and also their complements, are equiangular with the whole parallelogram; for each has an angle in common with it (152).

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

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