(ED+AE) (ED-AE) (II. 5, Cor.) AD DB, for ED AEED — EBDB. COR. 1.-If + refer to the 1st figure, and For, in the 1st figure, AC2 to the 2d, CD2 AD DB; therefore, adding to these equals CD2, AC2 = CD2 + AD·DB. And, in 2d fig. CD2 AC2 AD DB, and, adding to these equals AC2, and taking from them AD-DB; therefore AC2 CD2 AD.DB. COR. 2.-Half the difference of two unequal lines added to half their sum, gives the greater; and taken from half the sum, gives the less. For AD, DB (1st fig.) are two unequal lines, AE is half their sum, and ED half their difference; and AE + ED= AD, and AE-ED=DB. EXERCISES. 1. The square of one of the sides of a right-angled triangle is equal to the rectangle under the sum and difference of the hypotenuse and the other side. 2. The square of a perpendicular upon the hypotenuse of a right-angled triangle drawn from the opposite angle, is equal to the rectangle under the segments of the hypotenuse. 3. If a line be cut in medial section, the line composed of it and its greater segment is similarly divided. 4. If from any point lines be drawn to the angular points of a rectangle, the sums of the squares of those drawn to opposite angles are equal. 5. The sum of the squares of the sides of a quadrilateral is equal to the sum of the squares of its diagonals, and four times the square of the line joining their middle points. 6. The sum of the squares of two opposite sides of a quadrilateral, together with four times the square of the line joining their middle points, is equal to the sum of the squares of the other two sides and of the diagonals. 7. To bisect a given triangle by a line drawn from a point one of its sides. 8. The squares of the sum, and of the difference of two lines, are together double the squares of these lines. 9. To bisect a parallelogram by a line drawn from a point in one of its sides. 10. A line joining the middle points of two sides of a triangle, is parallel to the base, and equal to the half of it. 11. The quadrilateral formed by joining the successive middle points of the sides of a given quadrilateral, is a parallelogram. 12. The sum of the squares of the diagonals of a quadrilateral, is equal to twice the sum of the squares of the lines joining the middle points of the opposite sides. THIRD BOOK. DEFINITIONS. 1. An arc of a circle is any portion of the circumference. 2. The chord of an arc is a straight line joining its extremities. 3. The arc of a semicircle is called a semicircumference; and a radius is a semidiameter. 4. A segment of a circle is a figure contained by an arc and its chord. 5. An angle in a segment is an angle contained by two lines, drawn from any point in its arc, to the extremities of its chord. 6. An angle is said to insist or stand subtends it. upon the arc which 7. A sector of a circle is a figure contained by two radii and the intercepted arc. When the radii are perpendicular, the sector is called a quadrant. 8. Similar segments of a circle are those that contain equal angles. 9. Similar arcs of circles are those that subtend equal angles at the centre. 10. Similar sectors are those that are bounded by similar ares. (ED+AE) (ED — AE) (II. 5, Cor.) = AD DB, for ED AE ED - EBDB. COR. 1.-If+refer to the 1st figure, and to the 2d, CD2 AD DB; there fore, adding to these equals CD2, AC2 = CD2 + AD·DB. And, in 2d fig. CD2 · AC2 AD DB, and, adding to these equals AC2, and taking from them AD-DB; therefore AC2 CD2 AD.DB. COR. 2.-Half the difference of two unequal lines added to half their sum, gives the greater; and taken from half the sum, gives the less. For AD, DB (1st fig.) are two unequal lines, AE is half their sum, and ED half their difference; and AE + ED = AD, and AE-ED = DB. EXERCISES. 1. The square of one of the sides of a right-angled triangle is equal to the rectangle under the sum and difference of the hypotenuse and the other side. 2. The square of a perpendicular upon the hypotenuse of a right-angled triangle drawn from the opposite angle, is equal to the rectangle under the segments of the hypotenuse. 3. If a line be cut in medial section, the line composed of it and its greater segment is similarly divided. 4. If from any point lines be drawn to the angular points of a rectangle, the sums of the squares of those drawn to opposite angles are equal. 5. The sum of the squares of the sides of a quadrilateral is equal to the sum of the squares of its diagonals, and four times the square of the line joining their middle points. 6. The sum of the squares of two opposite sides of a quadrilateral, together with four times the square of the line joining their middle points, is equal to the sum of the squares of the other two sides and of the diagonals. 7. To bisect a given triangle by a line drawn from a point one of its sides. 8. The squares of the sum, and of the difference of two lines, are together double the squares of these lines. 9. To bisect a parallelogram by a line drawn from a point in one of its sides. 10. A line joining the middle points of two sides of a triangle, is parallel to the base, and equal to the half of it. 11. The quadrilateral formed by joining the successive middle points of the sides of a given quadrilateral, is a parallelogram. 12. The sum of the squares of the diagonals of a quadrilateral, is equal to twice the sum of the squares of the lines joining the middle points of the opposite sides. THIRD BOOK. DEFINITIONS. 1. An arc of a circle is any portion of the circumference. 2. The chord of an arc is a straight line joining its extremities. 3. The arc of a semicircle is called a semicircumference ; and a radius is a semidiameter. 4. A segment a circle is a figure contained by an arc and its chord. 5. An angle in a segment is an angle contained by two lines, drawn from any point in its arc, to the extremities of its chord. 6. An angle is said to insist or stand upon the arc which subtends it. 7. A sector of a circle is a figure contained by two radii and the intercepted arc. When the radii are perpendicular, the sector is called a quadrant. 8. Similar segments of a circle are those that contain equal angles. 9. Similar arcs of circles are those that subtend equal angles at the centre. 10. Similar sectors are those that are bounded by similar ares. 11. Equal circles are those that have equal radii; and concentric circles are those that have the same centre. 12. A straight line is said to touch a circle, or to be a tangent to it, when it meets the circle, and being produced does not cut it. 13. Circles are said to touch one another when they meet, but do not cut one another. Such circles may be called tangent circles. 14. The point in which a tangent and a circle, or two tangent circles, meet, is called the point of contact. 15. Chords are said to be equally distant from the centre of a circle, when the perpendiculars upon them, from the centre, are equal; and the chord on which the greater perpendicular falls, is said to be farther from the centre. 16. A secant is a straight line drawn from a point, without a circle, and meeting it in two points. A secant may be considered to be a chord produced, the point without the circle from which it is drawn being a point of external section. To find the centre of a given circle. Let ABC be the given circle; it is required to find its centre. Draw within it any chord AB, and bisect it in D (I. 10); from the point D draw DC at right angles to AB (I. 11), and produce it to E, and bisect CE in F. The point F is the centre of the circle ABC. For, if it be not, let, if possible, G be the centre, and join GA, GD, GB. Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG, are equal to the two BD, DG, each to each; and the radius GA is equal to the radius GB; therefore the angle ADG is equal to the angle GDB (I. 8); therefore (I. Def. 10) the angle GDB is a right angle. But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible; therefore G is not the centre of the circle ABC. In the same manner it can be shown, that no A D B |