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A SYLLABUS OF

hypotenuse is equal to the sum of the squares on the sides.

[Alternative proofs :(1) Euclid's. (2) By dividing two squares placed side by side into

parts, which may be combined so as to form a single square]

THEOR. 10. In an obtuse-angled triangle the square on the side

opposite the obtuse angle is greater than the squares on the other two sides by twice the rectangle contained by either side and the projection on it of the

other side. THEOR. II. In any triangle the square on the side opposite an

acute angle is less than the squares on the other two sides by twice the rectangle contained by either side

and the projection on it of the other side. COR. Conversely, the angle opposite a side of a triangle is

an acute angle, a right angle, or an obtuse angle, according as the square on that side is less than, equal to, or greater than, the sum of the squares on the

other two sides. THEOR. 12. The sum of the squares on two sides of a triangle is

double the sum of the squares on half the base and on the line joining the vertex to the middle point of the base.

THEOR, 13. If a straight line is divided internally or externally at

any point, the sum of the squares on the segments is double the sum of the squares on half the line and on the line between the point of division and the middle point of the line.

SECTION 2.

PROBLEMS.

PROB. 1. To construct a parallelogram equal to a given triangle

and having one of its angles equal to a given angle. PROB. 2. To construct a parallelogram on a given base equal

to a given triangle and having one of its angles equal

to a given angle. PROB. 3. To construct a parallelogram equal to a given recti

lineal figure and having one of its angles equal to a

given angle. PROB. 4. To construct a square equal to a given rectilineal

figure. PROB. 5. To construct a rectilineal figure equal to a given

rectilineal figure and having the number of its sides one less than that of the given figure; and thence to construct a triangle equal to a given rectilineal

figure. PROB. 6. To divide a straight line, either internally or ex

ternally, into two segments such that the rectangle contained by the given line and one of the segments may be equal to the square on the other segment.

BOOK III.

THE CIRCLE.

SECTION 1.

ELEMENTARY PROPERTIES.

For Definitions of a circle, its radius, and diameter, see Book I,

Definitions 8, 9, 10. For the notion of a circle regarded as a locus see Book I, LOCI

$ 2, i. DEF. 1. An arc is a part of a circumference. DEF. 2. A chord of a circle is the straight line joining any two

points on the circumference. When the arcs into which the chord divides the circumference are unequal, they are called the major and minor arcs respectively. Such arcs are said to be conjugate to

one another. DEF. 3. A segment of a circle is the figure contained by a

chord and either of the arcs into which the chord divides the circumference. The segments are called major or minor segments according as the arcs that

bound them are major or minor arcs. DeF. 4. The conjugate angles formed at the centre of a circle

by two radii are said to stand upon the conjugate arcs opposite them intercepted by the radii, the major

A SYLLABUS OF PLANE GEOMETRY.

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angle upon the major arc, and the minor angle upon

the minor arc. DEF. 5. A sector is the figure contained by an arc and the

radii drawn to its extremities. The angle of the sector is the angle at the centre which stands upon the arc of the sector.

DEF. 6. Circles that have a common centre are said to be

concentric.

The following properties of the circle are immediate

consequences of Book I, Def. 8: (a) A circle has only one centre. (6) A point is within, on, or without the circumference

of a circle, according as its distance from the centre

is less than, equal to, or greater than the radius. (c) The distance of a point from the centre of a circle

is less than, equal to, or greater than the radius, according as the point is within, on, or without the

circumference. THEOR. I. Circles of equal radii are identically equal. [By

Superposition.]
CoR. Two (different) circles whose circumferences meet one

another cannot be concentric. THEOR. 2. In the same circle, or in equal circles, equal angles

at the centre stand on equal arcs, and of two unequal angles at the centre the greater angle stands on the

greater arc. [By Superposition.] COR. 1. Sectors of the same, or of equal circles, which have

equal angles are equal, and of two such sectors which have unequal angles the greater is that which has the

greater angle. [By Superposition.] COR. 2. A diameter of a circle divides it into two equal parts,

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A SYLLABUS OF

and two diameters at right angles to one another

divide the circle into four equal parts. DEF. 7. The former are called semicircles; and the latter are

called quadrants. THEOR. 3. In the same circle, or in equal circles, equal arcs

subtend equal angles at the centre, and of two unequal arcs the greater subtends the greater angle at

the centre. [By Rule of Conversion.] COR. Equal sectors of the same, or of equal circles, have

equal angles, and of two unequal sectors the greater has the greater angle. [By Rule of Conversion.]

SECTION 2.

CHORDS.

THEOR. 4. In the same circle, or in equal circles, equal arcs are

subtended by equal chords; and of two unequal minor arcs the greater is subtended by the greater

chord. [By Theor. 3, and Book I, Theors. 5 and 14.] Cor. In the same circle, or in equal circles, of two unequal

major arcs the greater is subtended by the less

chord. THEOR. 5. In the same circle, or in equal circles, equal chords

subtend equal major and equal minor arcs; and of two unequal chords the greater subtends the greater minor arc and the less major arc. [By Rule of Con

version.] THEOR. 6. The straight line drawn from the centre to the middle

point of a chord is perpendicular to the chord. THEOR. 7. The straight line drawn from the centre perpendicular

to a chord bisects the chord.

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