$117.

320. Divide a given line into 3 equal parts, not using

321. that the ratio 2 : 3.

Draw a line through the vertex of a given so will be divided into two As which shall have the

322. Construct a right isosceles ▲ equivalent to a given square.

323. Find the locus of the middle points of all the chords in a given circle which can be drawn through a given point (1) in the circumference, (2) within the circle, (3) without the circle.

324. Given the line A B and the angle a. Construct largest possible that shall have A B for the base, an a for the vertical angle.

325. From a point without a circle draw a secant so that the intercepted chord shall subtend of the circumference.

326. From a point without a circle draw a secant so that the internal segment and the external segment shall be equal. Discuss Ex. 326.

327. Construct a triangle having given

(1) The base, altitude, and the median to the base.

Discuss.

Discuss.

(2) The angles and one median.
(3) One side, an ▲ adjacent to that side, and the
sum or the difference of the other two sides.
Discuss.

(4) The perimeter, one Z, and the altitude drawn
from the vertex of this . Discuss.

(5) The radius of the circumscribed circle, and the angles. Discuss.

BOOK V.

REGULAR POLYGONS.

MEASUREMENT OF THE CIRCLE.

328.

Recall your definition of a regular polygon. See § 104. The term regular polygon means a convex polygon unless otherwise stated.

What would you call an equilateral triangle? a square?

329.

Regular Convex Pentagon.

Regular Concave or Cross Polygon.

The subject of the regularity of polygons may be looked at from the standpoint of symmetry. By symmetry in Geomtry we mean that if a figure be turned half way round on a point as a pivot, each part of the figure will occupy the same space previously occupied by another part. If the figure, on being turned half way round, occupies its original position, it is said to have two-fold symmetry.

If an equilateral triangle be revolved about its center one-third of 360°, what will be its second position? Suppose it is turned two-thirds of 360° about the center, what will be its third position?

Discuss revolving a square about its center.

Is a right triangle symmetrical with regard to its center? an isosceles triangle?

If a figure be turned one-third of a revolution and it occupies its original position, the figure is said to have threefold symmetry.

What is four-fold symmetry? five-fold symmetry?

What is the symmetry with regard to a point illustrated by the following figures? Make figures illustrating other forms of symmetry.

For examples, observe wall paper and other decorations.

330.

From the standpoint of symmetry, polygons that are symmetrical are regular. Thus, a triangle is regular if it has three-fold symmetry. A heptagon is regular if it has sevenfold symmetry.

A polygon of n sides is regular if it has n-fold symmetry.

331.

By means of revolution show that a symmetrical octagon has (1) its sides equal, (2) its angles equal; (3) that a circle may be circumscribed about a regular polygon having the same center as the polygon; (4) that with the center of the polygon for center a circle may be inscribed within it.

From the special case just given, can you prove a general truth? State it.

Notice that we prove by symmetry (1) and (2), which are given as a definition in § 104.

332.

Into how many isosceles triangles may a regular triangle be divided by joining the center to the vertices? a regular quadrilateral? a regular pentagon? a regular hexagon?

PROPOSITION I.

Can you show that a regular polygon, P, of n sides, may be divided into n isosceles triangles ?

Write Prop. I.

333.

Cor. I. What do the bisectors of any two Zs of a regular polygon determine by their intersection?

334.

Cor. II. A point is equidistant from all the vertices of a regular polygon. How is it related to the sides of the polygon? Prcof.

335.

Cor. III. Show that a O may be circumscribed about or inscribed within a regular polygon and both Os have the same center.

The center in Cor. III. is called the center of a regular polygon and the radius of the circumscribing circle is called. the radius of a regular polygon; the radius of the inscribed O is called the apothem of a regular polygon. Draw figure and fix these terms in mind. What is meant by the at the center of a regular polygon?

336.

Cor. IV. If a regular polygon have n sides, show that each at the center = 4 rt. Zs÷n.

337.

PROPOSITION II.

Given: ABCDEF, an equilateral polygon inscribed in a circle. Can you prove it to be regular?

338.

Cor. I. If a circumference be divided into n equal arcs, (n > 2), what will the chords of these arcs form?