to four right angles, therefore, three angles of any regular polygon of more than six sides, must exceed four right angles.

Hence, no other regular figures exist for the purpose here required, except the equilateral triangle, the square, and the regular hexagon.

ISOPERIMETRY.

468. Theorem. Of all equivalent polygons of the same number of sides, the one having the least perimeter is regular.

Of several equivalent polygons, suppose AB and BC to be two adjacent sides of the one having the least perimeter. It is to be proved, first, that these sides are equal.

Join AC. Now, if AB

and BC were not equal,

B

C

there could be constructed on the base AC an isosceles triangle equivalent to ABC, whose sides would have less extent (395). Then, this new triangle, with the rest of the polygon, would be equivalent to the given polygon, and have a less perimeter, which is contrary to the hypothesis.

It follows that AB and BC must be equal. So of every two adjacent sides. Therefore, the polygon is equilateral.

It remains to be proved that the polygon will have all its angles equal.

Suppose AB, BC, and CD to be adjacent sides. Produce AB and CD till they meet at E. Now the triangle BCE is isosceles. For if EC, for example, were

longer than EB, we could then take EI equal to EB, and

EF equal to EC, and we

could join FI, making the two triangles EBC and EIF equal (284).

Then, the new polygon, having AFID for part of its perimeter, would be

A

B

equivalent and isoperimetrical to the given polygon having ABCD as part of its perimeter. But the given polygon has, by hypothesis, the least possible perimeter, and, as just proved, its sides AB, BC, and CD are equal.

If the new polygon has the same area and perimeter, its sides also, for the same reason, must be equal; that is, AF, FI, and ID. But this is absurd, for AF is less than AB, and ID is greater than CD. Therefore, the supposition that EC is greater than EB, which supposition led to this conclusion, is false. Hence, EB and EC must be equal.

Therefore, the angles EBC and ECB are equal (268), and their supplements ABC and BCD are equal. Thus, it may be shown that every two adjacent angles are equal.

It being proved that the polygon has its sides equal and its angles equal, it is regular.

469. Corollary.—Of all isoperimetrical polygons of the same number of sides, that which is regular has the greatest area.

470. Theorem. Of all regular equivalent polygons, that which has the greatest number of sides has the least perimeter.

It will be sufficient to demonstrate the principle, when one of the equivalent polygons has one side more than the other.

In the polygon having the less number of sides, join the vertex C to any point, as H, of the side BG. Then,

on CH construct an isosceles triangle, CKH, equivalent to CBH.

Then HK and KC are less than HB and BC; therefore, the perimeter GHKCDF is less than the perimeter of its equivalent polygon GBCDF. But the perimeter of the regular polygon AO is less than the perimeter of its equivalent irregular polygon of the same number of sides, GHKCDF (468). So much more is it less than the perimeter of GBCDF.

471. Corollary.—Of two regular isoperimetrical polygons, the greater is that which has the greater number of sides.

EXERCISES.

472.-1. Find the ratios between the side, the radius, and the apothem, of the regular polygons of three, four, five, six, and eight sides.

2. If from any point within a given regular polygon, perpendiculars be let fall on all the sides, the sum of these perpendiculars is a constant quantity.

3. If from all the vertices of a regular polygon, perpendiculars be let fall on a straight line which passes through its center, the Geom.-14

sum of the perpendiculars on one side of this line is equal to the sum of those on the other.

4. If a regular pentagon, hexagon, and decagon be inscribed in a circle, a triangle having its sides respectively equal to the sides of these three polygons will be right angled.

5. If two diagonals of a regular pentagon cut each other, each is divided in extreme and mean ratio.

6. Three houses are built with walls of the same aggregate length; the first in the shape of a square, the second of a rectangle, and the third of a regular octagon. Which has the greatest amount of room, and which the least?

7. Of all triangles having two sides respectively equal to two given lines, the greatest is that where the angle included between the given sides is a right angle.

8. In order to cover a pavement with equal blocks, in the shape of regular polygons of a given area, of what shape must they be that the entire extent of the lines between the blocks shall be a minimum.

9. All the diagonals being formed in a regular pentagon, the figure inclosed by them is a regular pentagon.

CHAPTER VIII.

CIRCLES.

473. The properties of the curve which bounds a circle, and of some straight lines connected with it, were discussed in a former chapter. Having now learned the properties of polygons, or rectilinear figures inclosing a plane surface, the student is prepared for the study of the circle as a figure inclosing a surface.

The circle is the only curvilinear figure treated of in Elementary Geometry. Its discussion will complete this portion of the work. The properties of other curves, such as the ellipse which is the figure of the orbits of the planets, are usually investigated by the application of algebra to geometry.

474. A SEGMENT of a circle is that portion cut off by a secant or a chord. Thus, ABC and CDE are seg

A SECTOR of a circle is that portion included between two radii and the arc intercepted by them. Thus, GHI is a sector.