By Jean-Pierre Petit
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Additional info for Les Aventures d'Anselme Lanturlu - Le Geometricon
29) convex polygon ABCDE, and polygonal line A B C D E F G A which surrounds it completely. We extend side AB in both directions until it intersects the surrounding polygon at M , N . By the preceding result, the length of the path AEDCB is less than AM B A G N B, and therefore the perimeter of AEDCBA is less than the perimeter of the polygon N M B A G N . This perimeter, in turn, is less than the surrounding line because the part M B A G N is common to both, and M N < M C D E F N . 28. Theorem.
The angle B can therefore be arbitrary (at least within certain limits). A quadrilateral which can be deformed under these conditions is called an articulated quadrilateral. This notion is important in practical applications of geometry. 1 47. Theorem. In a parallelogram, the two diagonals divide each other into equal parts. In parallelogram ABCD (Fig. 46), we draw diagonals AC, BD, which intersect at O. Triangles ABO, CDO are congruent because they have equal angles, and an equal side AB = CD (by the preceding theorem).
In rectangle ABCD (Fig. 47), the diagonals are equal because triangles ACD, BCD are congruent: they have side DC in common, ADC = DCB since they are both right angles, and AD = BC since they are opposite sides of a parallelogram. Corollary. In a right triangle, the median from the vertex of the right angle equals half the hypotenuse. This is true because if we draw parallels to the sides of the right angle through the endpoints of the hypotenuse, we form a rectangle, in which the median in question is half the diagonal.
Les Aventures d'Anselme Lanturlu - Le Geometricon by Jean-Pierre Petit