750. COROLLARY I. The chords joining the alternate vertices of a regular inscribed decagon form a regular inscribed pentagon.

751. COROLLARY II. Tangents drawn at the vertices of the regular inscribed pentagon and decagon form a regular circumscribed pentagon and a regular circumscribed decagon.

752. COROLLARY III. If the arcs are bisected and chords and tangents are drawn according to § 710, regular inscribed and circumscribed polygons of 20, 40, 80, etc., sides will be formed. The length of the side of a regular inscribed decagon

753. EXERCISE. is (√5-1)r.

754. EXERCISE. Find the length of a side of a regular inscribed pentagon. [In the R.A. A ADC (see the figure of § 749), AC is the side of the decagon, and AD is one half the difference between the radius and the side of the decagon.] √10-2√5

Ans.

r.

2

755. EXERCISE. Show that the sum of the squares described on the sides of a regular inscribed decagon and of a regular inscribed hexagon equals the square described on the side of a regular inscribed pentagon.

[Represent the sides of the pentagon, hexagon, and decagon by p, h, and d, respectively.

756. EXERCISE. What is the length of the side of a regular decagon inscribed in a circle having a diameter 4 in. long?

757. EXERCISE. If the side of a regular pentagon is 2√5 in., show that the radius of the circumscribed circle is √10 + 2√5 in.

PROPOSITION VIII. PROBLEM

758. To inscribe a regular pentedecagon in a circle.

A

B

Let O be the center of the given circle.

Required to inscribe a regular polygon of fifteen sides in the circle.

=

Lay off the chord AB side of regular inscribed hexagon, and the chord AC side of regular inscribed decagon.

=

The arc AB contains 60°, (?) and the arc AC, 36°. (?)

..the arc BC contains 24° and is of the circumference. The circumference can therefore be divided into fifteen parts, each equal to BC; and the chords joining the points of division form a regular inscribed pentedecagon.

Q.E.F.

759. COROLLARY I. Tangents drawn at the vertices of the inscribed pentedecagon form a regular circumscribed pentedecagon.

760. COROLLARY II. If the arcs are bisected, and chords and tangents are drawn as described in § 710, regular inscribed and circumscribed polygons of 30, 60, 120, etc., sides will be formed.

761. SCHOLIUM. In Propositions V., VI., VII., and VIII. we have seen that the circumference can be divided into the following numbers of equal parts:

The mathematician Gauss has shown that it is possible to divide the circumference into 2" +1 equal parts, n being a positive integer and 2" +1 a prime number.

It is therefore possible, by the use of ruler and compasses, to divide the circumference into 2, 3, 5, 17, 257, etc., equal parts.

[An elementary explanation of the division of the circumference into seventeen equal parts is given in Felix Klein's "Vorträge über ausgewählte Fragen der Elementar Geometrie."]

PROPOSITION IX. THEOREM

762. The arc of a circle is less than any line that envelops it and has the same extremities.

Let AMB be the arc of circle and ASB any other line enveloping it and passing through A and B.

Proof. Of all the lines (AMB, ASB, etc.) that can be drawn through A and B, and including the segment or area AMB, there must be one of minimum length.

ASB cannot be the minimum line, for draw the tangent CD to the arc AMB.

CD < CSD. (?)

ACDB ASB. (?)

The same can be shown of every other line (except AMB) passing through A and B and including the area AMB. .. the arc AMB is the minimum line.

Q.E.D.

763. COROLLARY I. The circumference of a circle is less than the perimeter of a circumscribed polygon and greater than the perimeter of an inscribed polygon.

PROPOSITION X. THEOREM

764. If the number of sides of a regular inscribed polygon is indefinitely increased, its apothem approaches the radius as a limit.

B

Let AB be the side of a regular inscribed polygon and oc be its apothem.

To Prove that oc approaches the radius as its limit when the number of sides is indefinitely increased.

AB consequently ap

By increasing the number of sides AB can be made as small as we please, but not equal to zero. proaches zero as a limit, and since 04 OC AB, OA - OC approaches zero as its limit; and oc approaches 04 as its limit.

Q.E.D.

765. COROLLARY. If the number of sides of a regular circumscribed polygon is indefinitely increased, the distance from a vertex to the center of the circle approaches the radius as a limit.

[Proof similar to § 764.]

PROPOSITION XI. THEOREM

766. If a regular polygon is inscribed in or circumscribed about a circle and the number of its sides is indefinitely increased,

I. The perimeter of the polygon approaches the circumference as its limit.

II. The area of the polygon approaches the area of the circle as its limit.

Let AB be the side of a regular circumscribed polygon, and CD (parallel to AB) be the side of a similar inscribed polygon.

I. To Prove that the perimeters of the polygons approach the circumference of the circle as a limit when the number of sides is indefinitely increased.

Proof. Draw OA, OB, and OE.

OA passes through C and OB through D. (?)

Let P and p stand for the perimeters of the circumscribed and inscribed polygons respectively.