9. The sum of the squares of two lines is equal to half the square of the sum, together with half the square of the difference. 10. ABC is a triangle, having the sides AB and AC equal: if AB is produced to D, so that BD is equal to AB, show that the square on CD is equal to the square on AB, together with twice the square on BC. 11. Show that in a parallelogram the squares of the diagonals are equal to the sum of the squares of all the sides. 12. The square on the base of an isosceles triangle, whose vertical angle is a right angle, is equal to four times the area of the triangle. 13. Any rectangle is the half of the rectangle contained by the diameters of the squares on its two sides. 14. If the points C, D, be equidistant from the extremities of the straight line AB, show that the squares constructed on AD and AC, exceed twice the rectangle AC, AD, by the square constructed on CD. 15. ABCD is a rectangle, E any point in BC, F in CD; show that the rectangle ABCD is equal to twice the triangle AEF, together with the rectangle BE, DF. 16. In any triangle ABC, if BP, CQ be drawn perpendicular to AC, AB, produced if necessary, then shall the square of BC be equal to the rectangle AB, BQ, together with the rectangle AC, CP. 17. If DE be drawn parallel to the base BC of an isosceles triangle ABC, then the square of BE is equal to the rectangle BC, DE, together with the square of CE. 18. If from any point within a rectangle lines be drawn to the angular points, the sums of the squares of those which are drawn to the opposite angles are equal. 19. If two sides of a trapezium be parallel to each other, the squares of its diagonals are together equal to the squares of its two sides, which are not parallel, and twice the rectangle contained by its parallel sides. 20. The squares of the diagonals of a trapezium are together double the squares of the two lines joining the bisections of the opposite sides. 21. ABC is a triangle in which C is a right angle, and DE is drawn from a point D in AC perpendicular to AB; show that the rectangle AB, AE is equal to the rectangle AC, AD. 22. In AB, the diameter of a circle, take two points C and D equally distant from the centre, and from any point E in the circumference draw EC, ED; show that the squares on EC and ED are together equal to the squares on AC and AD. 23. ABC is a triangle, of which the angle at C is obtuse, and the angle at B is half a right angle: D is the middle point of AB, and CE is drawn perpendicular to AB. Show that the square of AC is double of the squares of AD and DE. 24. Divide a given straight line into two parts, such that the squares of the whole line and of one of the parts shall be equal to twice the square of the other part. 25. In any triangle, if a line be drawn from the vertex bisecting the base, the sum of the squares of the two sides of the triangle double the sum of the squares of the bisecting line and of half the base. QUESTIONS AND EXERCISES ON BOOK III. DEFINITIONS. 1. What are equal circles? II. When is a straight line said to touch a circle? III. When are circles said to touch one another? IV. When are straight lines said to be equally distant from the centre of a circle? v. When is one straight line said to be farther from the centre of a circle than another? VI. What is a segment of a circle? VII. What is the angle of a segment ? VIII. What is an angle in a segment? IX. Upon what is an angle in a seginent of a circle said to insist? x. What is a sector of a circle? XI. What are similar segments of circles? PROPOSITIONS AND COROLLARIES OF BOOK III. Prop. 1. To find the centre of a given circle. Cor. If in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other. Prop. 2. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Prop. 3. If a straight line, drawn through the centre of a circle, bisect a straight line in it which does not pass through the centre, it shall cut it at right angles: and conversely, if it cut it at right angles, it shall bisect it. Prop. 4. If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other. Prop. 5. If two circles cut one another, they shall not have the same centre. Prop. 6. If one circle touch another internally, they shall not have the same centre. Prop. 7. If any point be taken in the diameter of a circle, which is not the centre, of all the straight lines which can be drawn from it to the circumnference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, of any others, that which is nearer to the line which passes through the centre is always greater than one more remote and from the same point there can be drawn only two equal straight lines to the circumference, one upon each side of the diameter. Prop. 8. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to that through the centre is always greater than one more remote: but of those which fall upon the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than one more remote: and only two equal straight lines can be drawn from the same point to the circumference, one upon each side of the line which passes through the centre. Prop. 9. If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. Prop. 10. One circumference of a circle cannot cut another in more than two points. Prop. 11. If one circle touch another internally in any point, the straight line which joins their centres, being produced, shall pass through that point of contact. Prop. 12. If two circles touch each other externally in any point, the straight line which joins their centres shall pass through that point of contact. Prop. 13. One circle cannot touch another in more points than one, whether it touch it on the inside or outside. Prop. 14. Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to one another. Prop. 15. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote: and the greater is nearer to the centre than the less. Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle; or, which is the same thing, no straight line can make so great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles to it, as not to cut the circle. Cor. The straight line which is drawn at right angles to the diameter of a circle from the extremity of it touches the circle; and it touches it only in one point; also there can be but one straight line which touches the circle in the same point. Prop. 17. To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. Prop. 18. If a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle. Prop. 19. If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line. Prop. 20. The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference. Prop. 21. The angles in the same segment of a circle are equal to one another. Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles. Prop. 23. Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another. Prop. 24. Similar segments of circles, upon equal straight lines, are equal to one another. Prop. 25. A segment of a circle being given, to describe the circle of which it is the segment. Prop. 26. In equal circles, equal angles stand upon equal circumferences, whether they be at the centres or circumferences. Prop. 27. In equal circles, the angles which stand upon equal circumferences are equal to one another, whether they be at the centres or circumferences. Prop. 28. In equal circles, equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less. Prop. 29. In equal circles, equal circumferences are subtended by equal straight lines. Prop. 30. To bisect a given circumference, that is, to divide it into two equal parts. Prop. 31. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. Cor. If the angle of a triangle be equal to the other two, it is a right angle, because the angle adjacent to it is equal to the same two, and when the adjacent angles are equal, they are right angles. Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle; the angles made by this line with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle. Prop. 33. Upon a given straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle. Prop. 34. From a given circle, to cut off a segment which shall contain an angle equal to a given rectilineal angle. Prop. 35. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Prop. 36. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it. Cor. If from any point without a circle there be drawn two straight lines cutting it, the rectangles contained by the whole lines and the parts of them without the circle are equal to one another. Prop. 37. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle. GEOMETRICAL EXERCISES ON BOOK III. 1. Describe a circle of given radius, which shall pass through two given points. 2. In the figure of Prop. III. 3, if any line be drawn parallel to AB, the diameter CD will bisect it. 3. Given two points in the circumference of a circle: describe the circle. 4. If two chords of a circle intersect each other, and make equal angles with the diameter passing through their point of contact, they are equal. 5. If one circle touch another internally, the distance between their centres is less than the difference of their radii. 6. Draw the shortest chord through a given point inside a circle. 7. If AD, CE be drawn perpendicular to the sides BC, AB of the triangle ABC, and DE be joined; prove that the angles ADE and ACE are equal to each other. 8. Parallel chords of a circle intercept equal arcs. 9. A quadrilateral is described so that its sides touch a circle: show that two of its sides are together equal to the other two sides. 10. Show that two tangents can be drawn to a circle from a given external point, and that they are of equal length. NOTE.-A tangent is a straight line which touches a circle. 11. Right-angled triangles are described on the same hypotenuse: show that the angular points opposite the hypotenuse all lie on a circle described on the hypotenuse as diameter. 12. A chord PAQ cuts the diameter of a circle in A in an angle which is half a right angle; show that the squares of AP and AQ are together double of the square of the radius. 13. A is any point in the diameter (or diameter produced) of a circle, whose centre is O, OB a radius perpendicular to the diameter; if AB cut the circle in P, and the tangent in P cut AO in C, show that AC=CP. 14. If two circles touch each other, and parallel diameters be drawn, then lines which join the extremities of these diameters will pass through the point of contact. 15. ABCD is a parallelogram; draw CE perpendicular to the diagonal BD, and show that the perpendiculars upon AB, AD, at the points B, D, will intersect in CE. 16. The exterior angle of a quadrilateral figure inscribed in a circle, is equal to the interior and opposite. 17. Two circles intersect in A, B, the centre of one being in the circumference of the other; draw any chord ACD cutting them both; show that CB=CD. 18. If AB, CD, be chords of a circle at right angles to each other, prove that the sum of the arcs AC, BD, is equal to the sum of the arcs AD, BC. 19. With a given radius to describe a circle touching two given circles. 20. Two circles intersect in the points A and B; through A and B any two straight lines CAD, EBF, are drawn cutting the circles in the points C, D, E, F; prove that CE is parallel to DF. |