45. If any number of parallelograms be constructed having their sides of given length, shew that the sum of the squares on the diagonals of each will be the same. 46. ABCD is a right-angled parallelogram, and AB is double of BC; on AB an equilateral triangle is constructed: shew that its area will be less than that of the parallelogram. 47. A point O is taken within a triangle ABC, such that the angles BOC, COA, AOB are equal; prove that the squares on BC, CA, AB are together equal to the rectangles contained by OB, OC; OC, OA; OA, OB; and twice the sum of the squares on OA, OB, OC. 48. If the sides of an equilateral and equiangular hexagon be produced to meet, the angles formed by these lines are together equal to four right angles. 49. ABC is a triangle right-angled at A; in the hypotenuse two points D, E are taken such that BD=BA and CE=CA; shew that the square on DE is equal to twice the rectangle contained by BE, CD. 50. Given one side of a rectangle which is equal in area to a given square, find the other side. 51. AB, AC are the two equal sides of an isosceles triangle ; from B, BD) is drawn perpendicular to AC, meeting it in D; shew that the square on BD is greater than the square on CD by twice the rectangle AD, CD. BOOK III. POSTULATE. A POINT is within, or without, a circle, according as its distance from the centre is less, or greater than, the radius of the circle. DEF. I. A straight line, as PQ, drawn so as to cut a circle ABCD, is called a SECANT. That such a line can only meet the circumference in two points may be shewn thus: Some point within the circle is the centre; let this be 0. Join OA. Then (Ex. 1, I. 16) we can draw one, and only one, straight line from O, to meet the straight line PQ, such that it shall be equal to OA. Let this line be OC. Then A and C are the only points in PQ, which are on the circumference of the circle. S E. II. DEF. II. The portion AC of the secant PQ, intercepted by the circle, is called a CHORD. DEF. III. The two portions, into which a chord divides the circumference, as ABC and ADC, are called ARCS. DEF. IV. The two figures into which a chord divides the circle, as ABC and ADC, that is, the figures, of which the boundaries are respectively the arc ABC and the chord AC, and the arc ADC and the chord AC, are called SEGMENTS of the circle. DEF. V. The figure AOCD, whose boundaries are two radii and the arc intercepted by them, is called a SECTOR. DEF. VI. A circle is said to be described about a rectilinear figure, when the circumference passes through each of the angular points of the figure. 1 the figure is said to be inscribed in the circle. PROPOSITION I. THEOREM. The line, which bisects a chord of a circle at right angles, must contain the centre. Let the st. line CE bisect the chord AB at rt. angles in D. Then the centre of the must lie in CE. For if not, let O, a pt. out of CE, be the centre ; and join OA, OD, OB. Then, in AS ODA, ODB, ·· AD=BD, and DO is common, and OA = OB ; and.. 4 ODB is a right . But CDB is a right, by construction; L .. 4 ODB = CDB, which is impossible; .. O is not the centre. I. c. I. Def. 9 Thus it may be shewn that no point, out of CE, can be the centre, and.. the centre must lie in CE. COR. If the chord CE be bisected in F, then F is the centre of the circle. PROPOSITION II. THEOREM. If any two points be taken in the circumference of a circle, the straight line, which joins them, must fall within the circle. Let A and B be any two pts. in the Oce of the O ABC. Then must the st. line AB fall within the ©. .. the distance of D from O is less than the radius of the O, and .. D lies within the O. And the same may be shewn of any other pt. in AB. .. AB lies entirely within the O. Post. |