PROBLEM III.

314. To describe an isosceles triangle having each of the angles at the base double the angle at the vertex.

CONSTRUCTION. Take any sect, AB, and, by 313, divide it in C so that rectangle AB,AC BC. With center C and radius CB, describe a circle.

=

With same radius, but center A, describe a circle intersecting the preceding circle in D. Join AD, BD, CD.

(306. The square on a side opposite an acute angle is less than the squares on the other two sides by twice the rectangle of either and the projection of the other on it.)

But AC

2AF,

:. AB,AC = AB,2AF = 2AB,AF;

.. by our initial construction, BC2 = 2AB,AF.
:. AB3 +AD = BD + BC2.

But, by construction, AD = BC, :. AB3 = BD,

:. AB = BD,

ADB = 4 DAB = ACD = 4 B + ¥ BDC = 2 ¥ B.

(173. The exterior angle of a triangle equals the sum of the two opposite interior

angles.)

BOOK III.

THE CIRCLE.

I. Primary Properties.

315. If a sect turns about one of its end points, the other end point describes a curve called the Circle.

316. The fixed end point is called the Center of the circle. 317. The moving sect in any position is called a Radius of the circle.

318. As the motion of a sect does not enlarge or diminish it, all radii are equal.

319. Since the moving sect, after revolving through a perigon, returns to its original position, therefore the moving end point describes a closed curve.

This divides the plane into two surfaces, one of which is swept over by the moving sect. This finite plane surface is

called the surface of the circle.

Any part of the circle is called an Arc.

THEOREM I.

320. The sect to a point from the center of a circle is less than, equal to, or greater than, the radius, according as the point is within, on, or without the circle.

A

PROOF. If a point is on the circle, the sect drawn to it from the center is a radius, for it is one of the positions of the describing sect. Any point, Q, within the circle lies on some radius, OQR,

If S is without the circle, then the sect OS contains a radius OR,

OS > OR.

321. By 33, Rule of Inversion, a point is within, on, or without the circle according as its sect from the center is less than, equal to, or greater than, the radius.

322. A Secant is a line which passes through two points on the circle.

THEOREM II.

323. A secant can meet the circle in only two points.

PROOF. By definition, all sects joining the center to points on the circle are equal, but from a point to a line there can be only two equal sects.

(155. No more than two equal sects can be drawn from a point to a line.)

324. A Chord is the part of a secant between the two points where it intersects the circle.

325. A Segment of a circle is the figure made by a chord and one of the two arcs into which the chord divides the circle.

326. When two arcs together make

329. A segment is called a Major or Minor Segment according as its arc is a major or minor arc.