NOTE.

2.

E

D

B

ΤΑ

OCB and CBO <rt. 4, ... each = oblique . 3. .. CO and BO cannot be ||, and ... meet as at O.

The inscription and circumscription of regular polygons is seen to depend upon the partition of the perigon. Elementary geometry is thus limited to the inscription and circumscription of regular polygons of 2", 3.2", 5.2", 15.2′′ sides; or, since the discovery by Gauss, to polygons the number of whose sides is represented by the product of 2" and one or more different prime numbers of the form 2m + 1.

In addition to regular convex polygons, cross polygons can also be regular, the common five-pointed star being an example.

EXERCISES. 465. Solve pr. 7 by bisecting the sides AB, BC by perpendiculars, thus determining O.

466. Inscribe a regular cross pentagon in a circle. (The regular cross pentagon, the pentagram, was the badge of the Pythagorean school.) 467. The distance from the center to a side of the inscribed equilateral triangle equals r/2.

468. The area of an inscribed equilateral triangle is half that of a regular hexagon inscribed in the same circle.

Problem 8. To inscribe a circle in a given regular

2. OA bisects WX, ... A lies between W and X, and so

for B, C,....

3. ... WX = XY = ·····, .'. OA = OB

= .....

III, th. 5

III, th. 7

4. if with center O and radius OA a ◇ is described,

then WX, XY,..... will be tangent to the O.

5... the is inscr. in the polygon.

III, th. 9, cor. 3

Def. inscr. O

COROLLARIES. 1. The inscribed and circumscribed circles of

a regular polygon are concentric.

2. The bisectors of the angles of a regular polygon meet in the common in- and circumcenter.

For by the proof of pr. 7 they meet in O, and by cor. 1 O is the common in- and circumcenter.

3. The perpendicular bisectors of the sides of a regular polygon meet in the common in- and circumcenter. (Why?)

DEFINITIONS. The radius of the circumscribed circle is called the radius of a regular polygon; the radius of the inscribed circle, the apothem of that polygon; the common center of the two circles, the center of that polygon.

EXERCISE. 469. Draw a diameter AB of a circle with center O; then with center A and radius AO draw an arc cutting the circumference in C, D; draw CD, DB, BC, and prove ▲ BCD equilateral.

Theorem 10. The area of a regular polygon equals half the product of the apothem and perimeter.

Given an inscribed regular polygon,

of area a, perimeter p, apothem m.

2. Let t be one of the A formed

by joining O to two consecu

tive vertices, and s a side of the polygon.

S

m

COROLLARIES. 1. The areas of regular polygons of the same number of sides are proportional to the squares of their apothems, of their radii, or of their sides.

2. The perimeters of regular polygons of the same number of sides are proportional to their apothems, their radii, or their sides.

Proved with cor. 1.

EXERCISES. 470. The area of an inscribed regular hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

471. Show how, with compasses alone, to divide a circumference into six equal arcs.

472. Prove that if AB, CD, two diameters of a circle, are perpendicular to each other, then ACBD is an inscribed square.

=

473. Let OX be the perpendicular bisector of line-segment AB at 0; lay off on OX, OD = AO; and, on DX, lay off DC DB; then prove that C is the center of the circumscribed about the regular octagon of which AB is a side.

Section 4. The Mensuration of the Circle.

POSTULATE OF LIMITS. The circle and its circumference are the respective limits which the inscribed and circumscribed regular polygons and their perimeters approach if the number of their sides increases indefinitely.

This statement is so evident that a proof

is not considered necessary. Like valid proofs of many fundamental principles, it is too difficult for an elementary text-book. The following may be profitably read by the student in connection with the postulate : 1. In the figure, suppose a portion of an in- and circumscribed regular n-gon represented.

2. Then each exterior angle equals in each figure.

3... each interior angle equals 180°.

P

a M

360°

Α'

M'

B'

360°

n

7. .. the inscribed polygon the circle, and its perimeter ± the circumference.

Similarly for the circumscribed polygon.

COROLLARIES. 1. The circumscribed regular polygon and its perimeter are respectively greater than the circle and its circumference; the inscribed, and its perimeter, less.

2. If, on any finite closed curve, n points are assumed equidistant from each other, and each connected with the succeeding point by a straight line, then the curve is the limit

which the broken line approaches if n increases indefinitely.

Theorem 11. The ratio of the circumference to the diameter of a circle is constant.

Proof. 1. Suppose any two circles, of circumferences c, c', radii r, r', and diameters d, d', respectively, to have similar regular polygons inscribed in them, of perimeters p, p', respectively.

NOTE.

2. Then pp'=r: r'.

Th. 10, cor. 2

=2r2r'd: d'. IV, fund. prop. VIII 3. And ·.· r, r', d, d' do not change when the number of sides of the polygons is doubled, quadrupled, ..... Def. radius polyg.

4. And ... pc, and p' = c',

5.

... c: c' = d: d'.

6.

Post. of limits

Th. of limits

..cd= c': d'= the same for any .

IV, fund. prop. III

This constant ratio c d is designated by the symbol π (pi), the initial letter of the Greek word for circumference (periphereia). The value of π is discussed in th. 13.

2. If the radius of a circle is 1, then c = 2π, or a semicircumference equals π.

EXERCISES. 474. Find, in terms of the radius of the circle, r, the side, apothem, and area of the inscribed and circumscribed equilateral triangle.

475. Also of the inscribed and circumscribed square.

476. The diagonals of a regular pentagon cut each other in extreme and mean ratio.

477. If ABCDE is a regular pentagon, and AD cuts BE at P, prove that AP AE AE: AD.

478. To construct a regular pentagon equal to the sum of two given regular pentagons.

=

479. If d side of an inscribed regular decagon, p = that of an inscribed regular pentagon, r = radius of the circle, prove that p2 = d2 + r2.