24 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. 27 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.] 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, 28 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. |