28 II. ON TRIANGLES
The path ACDB is shorter than ACHB because they have a common portion
ACD, and the remainder BD of the first is less than the remainder DHB of
the second. In turn, the path ACHB is shorter than AGD E F B because, after
removing the common parts AC, HB, the segment CH which is left is smaller
than the broken line CGD E F H. In the same way, AGD E F B is less than
AC
D E F B, because AG is less than AC G. Thus
ACDB ACHB ACGD E F B AC D E F B.
QED
Corollary. The perimeter of a convex polygon is less than the perimeter of
any closed polygonal line surrounding it completely.
Figure 29
Consider (Fig. 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 AMB A G NB, and therefore
the perimeter of AEDCBA is less than the perimeter of the polygon NMB A G N.
This perimeter, in turn, is less than the surrounding line because the part MB A G N
is common to both, and MN MC D E F N.
28. Theorem. If two triangles have a pair of unequal angles contained by two
sides equal in pairs, and the third sides of the triangles are unequal, then the greater
side is opposite the greater angle.
Figure 30
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