401. COR. 2. By drawing tangents at A, B, C, and D, a square may be circumscribed about the circle.

402. COR. 3. If R is the radius of a circle, the side of the inscribed square equals R√2.

Ex. 1291. To circumscribe a regular octagon about a given circle. Ex. 1292. To construct a regular octagon, having a given side. Ex. 1293. Find the area of a square, if its radius is equal to r.

PROPOSITION V. PROBLEM

403. To inscribe a regular hexagon in a given circle.

B

Construction. In the given circle ACD, draw the radius 40. From 4 as a center, with a radius equal to 04, draw an arc meeting the circle in B. Draw AB.

[To be completed by the student.]

404. COR. 1. By joining the alternate vertices of an inscribed regular hexagon, an inscribed equilateral triangle is formed.

405. COR. 2. Regular polygons of 3, 6, 12, 24, etc., sides may be inscribed in and circumscribed about a given circle.

Ex. 1295. To circumscribe a regular hexagon about a given circle. Ex. 1296. To construct a regular polygon of twelve sides, having given a side.

Ex. 1297. Ex. 1298. its radius.

Ex. 1299.

Find the area of a regular hexagon if its radius is equal to r.
The apothem of an equilateral triangle is equal to one half

The area of an inscribed equilateral triangle is equal to one

half the area of the inscribed regular hexagon.

Ex. 1300. The areas of triangles inscribed in equal circles are to each other as the products of their three sides. (339.)

Ex. 1301. A square constructed on a diameter of a circle is equivalent to twice the area of the inscribed square.

407. DEF. A straight line is said to be divided in extreme and mean ratio when it is divided into two segments, such that the greater is the mean proportional between the smaller and the whole line.

Thus, AB is divided by C in A

extreme and mean ratio, if

AB: AC AC: CB.

PROPOSITION VI. PROBLEM

408. To divide a line in extreme and mean ratio.

B

Required to divide a in extreme and mean ratio.

Analysis. Suppose AC, or x, was the greater of the required

The proof of this construction is contained in the analysis. If we wish to write it out in detail, we may copy the last six

lines of the analysis in reverse order, and assign reasons for each step.

409. NOTE. The method used for the discovery of the preceding problem is based upon algebra. Hence, it is usually called algebraic analysis. For further details of this method see appendix.

PROPOSITION VII. PROBLEM

410. To construct the side of a regular decagon inscribed in a given circle.

Given 0.

Required to construct the side of a regular inscribed decagon.
Construction. Divide a radius 04 in extreme and mean ratio.
Draw chord 4D equal to OC, the greater segment of OA.
AD is the side of the required decagon.

But A OAD being isosceles, the similar triangle DAC must be

411. COR. 1. By joining the alternate vertices A, E, G, etc. an inscribed regular pentagon is formed.

412. COR. 2. Regular polygons of 5, 10, 20, etc., sides may be inscribed in and circumscribed about a given circle.

PROPOSITION VIII. PROBLEM

413. To construct the side of a regular polygon of fifteen sides inscribed in a given circle.

B