34. If BD be drawn bisecting AC, one of the sides of the triangle ABC, in D, and AE be drawn perpendicular to the base, shew that the square upon BD is equal to the sum or difference of the square upon the half of BD and the rectangle BC, BE, according as E lies in BC or in BC produced. 35. Any rectangle is half the rectangle contained by the diameters of the squares upon its two sides. 36. If from any point within a rectangle lines be drawn to the angular points, the sums of the squares upon those drawn to the opposite angles will be equal. 37. The squares of the diagonals of a parallelogram are together equal to the squares of the four sides. 38. The squares of the diagonals of a quadrilateral are together less than the squares of the four sides by four times the square of the line joining the bisections of the diagonals. 39. The squares of the diagonals of any quadrilateral are together double of the squares of the two lines joining the bisections of the opposite sides. 40. The squares of the sides of any triangle are together triple of the squares of the distances of the angles from the point of intersection of lines drawn from them to the bisections of the opposite sides. 41. If two opposite sides of any quadrilateral be bisected, the sum of the squares of the other two sides, together with the squares of the diagonals, is equal to the sum of the squares of the sides bisected together with four times the square of the line joining the points of section. 42. If DE be drawn parallel to the base BC of an isosceles triangle ABC, then the square of BE is equal to the rectangle of BC, CD, together with the square of CE. 43. The squares of the diagonals of a trapezium are together equal to the squares of its two parallel sides, with twice the rectangle contained by its parallel sides. 44. If BD, CE be squares described upon the sides AB, AC, of any triangle, shew that the squares of BC and DE are together double of the squares of AB and AC. 45. If squares be described on the three sides of any triangle, and the angular points of the squares be joined, the sum of the squares of the sides of the hexagonal figure thus formed will be equal to four times the sum of the squares of the sides of the triangle. 46. If two points be taken in the diameter of a circle equally distant from the centre, the sum of the squares of two lines drawn from these points to any point in the circumference will be constant. 47. The hypothenuse AB of a right-angled triangle ABC is trisected in the points D, E: shew that, if CD, CE be joined, the sum of the squares on the sides of the triangle CDE is equal to two-thirds of the square on AB. 48. Divide a given line into two parts, so that their rectangle may be equal to a given square. 49. If the areas of a triangle and of a square be equal, the perimeter of the triangle will be the greater. 50. ABCD is a quadrilateral, E the middle point of the line joining the bisections of the diagonals; if with E as centre any circle be described, shew that for every point P in this circle, PA2+ PB2+ PC2+ PD2 is constant, and equals EA2+EB2+EC2+ED2+4EP2. BOOK III. DEFINITIONS. 1. EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal. This is not a definition, but a theorem, the truth of which is evident; for, if the circles be applied to one another so that their centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal. O II. A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it. III. Circles are said to touch one another, which meet but do not cut one another. IV. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal: v. And the straight line on which the greater perpendicular falls, is said to be farther from the centre. VI. A segment of a circle is the figure contained by a straight line and the circumference it cuts off. VII. The angle of a segment is that which is contained by the straight line and the circumference. VIII. An angle in a segment is the angle contained by two straight lines drawn from any point in the cir cumference of the segment to the extremities of the straight line which is the base of the segment: IX. And an angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle. x. A sector of a circle is the figure contained by two straight lines drawn from the centre, and the circumference between them. XI. Similar segments of circles are those in which the angles are equal, or which contain equal angles. PROP. I. PROB. 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 straight line AB and bisect it in D; from the point D draw DC at right angles to AB, and produce CD to E, and bisect CE in F: the point F shall be the centre of the circle ABC. For, if the centre be in CE, it is plain that it must be F, the middle point of CE: but if it be not in CE, let, if possible, G be the centre, and join GA, GD, GB: Then, because AD is equal to BD, 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 base AG is equal to the base BG, because they are both drawn from the centre G to the circumference-therefore the angle ADG is equal to the angle BDG: But when a straight line, standing upon another straight line, makes the adjacent FG D B E angles equal to one another, each of these angles is a right angle; therefore the angle BDG is a right angle: But BDF is also a right angle; therefore the angle BDF is equal to the angle BDG, the greater to the less-which is absurd: therefore G is not the centre of the circle ABC: And in like manner it can be shewn that no other point but F is the centre; that is, F has been found, the centre of the circle ABC. Q. E. F. COR. From this it is manifest that, if in a circle one straight line bisect another at right angles, the centre of the circle is in the line which bisects the other. PROP. II. THEOR. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference: the straight line drawn from A to B shall fall within the circle. C For if not, let it fall, if possible, without, as AEB: find D the centre of the circle ABC; and join DA, DB; in the circumference AB take any point F, join DF, and produce it to E: Then, because DA is equal to DB, the angle DAB is equal to the angle DBA: And because AE, a side of the triangle DAE, is produced to B, the exterior angle DEB is greater than the angle DAE: But DAE was proved equal to the angle DBE; therefore the angle DEB is A 'D F E B N. B. Whenever the expression "straight lines from the centre," or "drawn from the centre," occurs, it is to be understood that they are drawn to the circumference. |