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CFG is a right angle by construction, therefore ADG is a right angle, wherefore CD is perpendicular to AB.

EXERCISES. 1. If a line drawn from the centre of a circle bisect or be perpendicular to a chord, it will bisect and be perpendicular to all chords that are parallel to the former.

2. If two circles cut each other, the line joining the points of intersection is bisected perpendicularly by the line joining their centres.

3. If two circles cut one another, and from one of the points of intersection two diameters be drawn, their other extremities, and the other point of intersection, will be in one straight line.

4. A line that bisects two parallel chords in a circle, is also perpendicular to them.

5. Two concentric circles intercept between them, two equal portions of a line cutting them both.

6. Of these five conditions-of passing through the centre of a circle, bisecting a chord, being perpendicular to a chord, bisecting the subtended angle at the centre, bisecting the subtended arc of the chord—if a line fulfil any two, it will also fulfil the other three.

This exercise comprehends ten theorems, of which three are proved in III. 3, and its corollary.

7. If a circle be described on the radius of another circle, any chord in the latter, drawn from the point in which the circles meet, is bisected by the former.

8. The exterior angle of a quadrilateral figure inscribed in a circle, is equal to the interior and opposite.

9. Parallel chords in a circle intercept equal arcs.

10. If two circles touch one another, either internally or externally, the chords of the two arcs, intercepted by two lines drawn through the point of contact, are parallel.

11. If a tangent to a circle be parallel to a chord, the point of contact bisects the intercepted arc.

12. If a chord to the greater of two concentric circles be a tangent to the less, it is bisected in the point of contact.

13. If any number of circles intersect a given circle, and pass through two given points, the lines joining the intersections of each circle will all meet in the same point.

14. Through two given points, to describe a circle touching a given circle.

15. If two chords in a circle intersect each other perpendicularly, the sum of the squares on their four segments is equal to the square on the diameter.

16. Perpendiculars, from the extremities of a diameter of a circle, upon any chord, cut off equal segments.

17. Given the vertical angle, the base, and the altitude of a triangle, to construct it.

18. In a circle, the sum of the squares on two lines drawn from the extremities of a chord, to any point in a diameter parallel to it, is equal to the sum of the squares on the segments of the diameter.

19. Given the vertical angle, the base, and the sum of the sides of a triangle, to construct it.

20. If the points of contact of two tangents to a circle be joined, any secant drawn from their intersection is divided into three segments, such that the rectangle under the secant and its middle segment is equal to that under its extreme segments.

21. If the opposite sides of a quadrilateral inscribed in a circle be produced to meet, the square on the line joining the points of concourse, is equal to the sum of the squares on the two tangents from these points.

22. Two parallel chords in a circle are respectively six and eight inches in length, and are one inch apart; find the diameter of the circle.

23. If two circles cut each other, the greatest line that can be drawn through either of the points of intersection, and is terminated by the circles, is parallel to the line joining their centres.

24. Describe a circle which shall touch a given circle in a given point, and also touch a given straight line.

25. Of all lines which touch the interior, and are bounded by the exterior of two circles which touch internally, the radius of the inner circle being greater than half that of the outer, the greatest is that which is parallel to the common tangent.

26. Shew that the two tangents to a circle drawn from the same point without it are equal to one another; and hence prove that the sums of the opposite sides of any quadrilateral described about a circle are equal, and the angles subtended at the centre of the circle by any two opposite sides are together equal to two right angles.

27. Describe a circle whose centre shall be in the base of a right-angled triangle, and which shall touch the hypotenuse and pass through the right angle.

28. If a circle be described about an equilateral triangle, and any point be taken in the circumference, the sum of the lines drawn from that point to the two adjacent angles, shall be equal to the line drawn from the same point to the opposite angle.

29. Determine a point in the diameter of a circle produced, from which a tangent being drawn to the circle, it shall be equal to the diameter.

30. Describe a circle which shall touch a given circle, and also touch a given line in a given point.

31. If in the diameter or diameter produced of a circle, two points be taken equally distant from the centre; the sum of the squares on the distances of any point on the circumference from these two points is constant. Also shew that if the point taken on the circle be in the diameter, it gives the proof of the ninth or tenth proposition of the Second Book of Euclid, according as the points are in the diameter produced, or in the diameter.

32. If on the three sides of any triangle, equilateral triangles be described; straight lines joining the centres of the circles described about these three triangles, will form an equilateral triangle.

FOURTH BOO K.

DEFINITIONS.

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* 1. A rectilineal figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is : inscribed, each upon each. 62. In like manner, a figure is said to be described about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each. 1.3. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.

4. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle.

5. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure.

6. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described.

7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

8. A regular polygon has all its sides equal, and also all its angles.

9. A polygon of five sides is called a pentagon; of six, a hexagon; of seven, a heptagon ; of eight, an octagon; of nine, a nonagon ; of ten, a decagon; of eleven, an undecagon; of twelve, a dodecagon; and of fifteen, a quindecagon or pentedecagon.

10. The centre of a regular polygon is a point equally distant from its sides or angular points.

11. The apothem of a regular polygon is a perpendicular from its centre upon any of its sides.

12. The perimeter or periphery of any figure is its circumference or whole boundary.

PROPOSITION I. PROBLEM. In a given circle to place a straight line, equal to a given straight line not greater than the diameter of the circle.

Given the circle ABC, and the straight line D, not greater than the diameter of the circle; it is required to place in the circle ABC a straight line equal to D.

(Const.) Draw BC a diameter of the circle ABC; then, if BC is equal to D, the thing required is done; for in the circle ABC a straight line BC is placed equal to D; but if it is not, BC is greater than D; make CE equal to D (I.3);

and from the centre C, at the distance CE, describe the circle AEF, and join CA; (Dem.) therefore, because C is the centre of the circle AEF, CA is equal to CE; but D is D- E equal to CE; therefore D is equal. to CA. Wherefore, in the circle ABC, a straight line is placed, equal to the given straight line D, which is not greater than the diameter of the circle.

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PROPOSITION II. PROBLEM. In a given circle to inscribe a triangle equiangular to a given triangle.

Given the circle ABC, and the triangle DEF; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.

(Const.) Draw the straight line GAH touching the circle in the

D / point A (III. 17), and at the point A, in the straight line AH, make the angle HAC equal to the angle DEF (1.23); and at the E point A, in the straight line AG, make the angle GAB equal to the angle DFE, and join BC. (Dem.) Therefore, because HAG touches the circle ABC, and AC is drawn from the point of contact, the angle HAC is equal to the angle ABC in the

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