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three given circles, none of which lies within the other, the tangents so drawn shall be equal.

58. Describe three circles touching each other and having their centers at three given points. In how many different ways may this be done?

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59. Describe in a given circle three circles which shall touch one another and the given circle.

60. Find the center and diameter of a circle that touches three given circles, each of which touches the other two.

61. From three given points as centers, describe three circles each of which touches the other two. In how many ways may this be done? Find also the center of the circle which passes through the points of contact.

62. If three circles touch each other in any manner, the tangents at the points of contact pass through the same point.

63. Given three unequal circles which do not intersect, and let pairs of double tangents be drawn internally to each pair of them, the three intersections will be in one right line.

64. The centers of three circles (A, B, C,) are in the same straight line, B and C touch each other externally and A internally, if a line be drawn through the point of contact of B and C, making any angle with the common diameter, then the portion of this line intercepted between C and A, is equal to the portion intercepted between B and A.

65. - A, B, C, are three given points, find the position of a circle such that all the tangents to it drawn from the points A, B, C shall be equal to one another. What is that circle which is the superior limit to those that satisfy the above condition?

66. A, B, C, are three given points in the same plane, but not in the same straight line, determine the center and the position of a circle, such that three tangents AP, AQ, AR, drawn from the points A, B, C, shall be respectively equal to three given straight lines.

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67. The straight line AB joining A, B, the centers of two circles, whose radii are R, r respectively, is divided in C, so that AC2-BO2 R2 - r2, and a straight line is drawn from C perpendicular to AB; prove that the tangents drawn to both circles from any point in this Îine are equal.

VII.

68. Draw a straight line which shall touch two given circles; (1) on the same side; (2) on the alternate sides.

69. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centers.

70. Let C, C' be the centers of two circles, draw two lines touching them on the same side in A, A', and on opposite sides in B, B', then AA" - BB12 = 4CA. C'A'.

71.

Find the point in the line joining the centers of two circles which do not meet, from which the tangents drawn to the two circles are equal.

GEOMETRICAL EXERCISES ON BOOK XI.

THEOREM I.

If a straight line be perpendicular to a plane, its projection on any other plane, produced if necessary, will cut the common intersection of the two planes at right angles.

Let AB be any plane,

and CEF another plane intersecting the former at any angle in the line EF;

and let the line GH bé perpendicular to the plane CEF.

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Draw GK, HL perpendicular on the plane AB,
and join LK,

then LK is the projection of the line GH on the plane AB;
produce EF, to meet KL in the point L;

then EF, the intersection of the two planes, is perpendicular to LK, the projection of the line GH on the plane AB.

Because the line GH is perpendicular to the plane CEF,

every plane passing through GH, and therefore the projecting plane GHKL is perpendicular to the plane CEF;

but the projecting plane GHLK is perpendicular to the plane AB; (constr.)

hence the planes CEF, and AB are each perpendicular to the third plane GHLK;

therefore EF, the intersection of the planes AB, CEF, is perpendicular to that plane;

and consequently, EF is perpendicular to every straight line which meets it in that plane;

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but EF meets LK in that plane.

Wherefore, EF is perpendicular to KL, the projection of GH on the plane AB.

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THEOREM II.

Prove that four times the square described upon the diagonal of a rectangular parallelopiped, is equal to the sum of the squares described on the diagonals of the parallelograms containing the parallelopiped.

Let AD be any rectangular parallelopiped; and AD, BG two diagonals intersecting one another; also AG, BD, the diagonals of the two opposite faces HF, CE.

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Then it may be shewn that the diagonals AD, BG, are equal; as also the diagonals which join CF and HE: and that the four diagonals of the parallelopiped are equal to one another.

The diagonals AG, BD of the two opposite faces HF, CE are equal to one another: also the diagonals of the remaining pairs of the opposite faces are respectively equal.

And since AB is perpendicular to the plane CE, it is perpendicular to every straight line which meets it in that plane,

therefore AB is perpendicular to BD,

and consequently ABD is a right-angled triangle.
Similarly, GDB is a right-angled triangle.

And the square on AD is equal to the squares on AB, BD, (1. 47.) also the square on BD is equal to the squares on BC, CD, therefore the square on AD is equal to the squares on AB, BC, CD; similarly the square on BG or on AD is equal to the squares on AB, BC, CD.

Wherefore the squares on AD and BG, or twice the square on AD, is equal to the squares on AB, BC, CD, AB, BC, CD;

but the squares on BC, CD are equal to the square on BD, the diagonal of the face CE;

similarly, the squares on AB, BC are equal to the square on the diagonal of the face HB;

also the squares on AB, CD, are equal to the square on the diagonal of the face BF; for CD is equal to BE.

Hence, double the square on AD is equal to the sum of the squares on the diagonals of the three faces HF, HB, BC.

In a similar manner, it may be shewn, that double the square on the diagonal is equal to the sums of the squares on the diagonals of the three faces opposite to HF, HB, BC.

Wherefore, four times the square on the diagonal of the parallelopiped is equal to the sum of the squares on the diagonals of the six faces.

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3. If two straight lines are parallel, the common section of any two planes passing through them is parallel to either.

4. If two straight lines be parallel, and one of them be inclined at any angle to a plane; the other also shall be inclined at the same angle to the same plane.

5. Parallel planes are cut by parallel straight lines at the same angle.

6. If two straight lines in space be parallel, their projections on any plane will be parallel.

7. Shew that if two planes which are not parallel be cut by two other parallel planes, the lines of section of the first by the last two will contain equal angles.

8. If four straight lines in two parallel planes be drawn, two from one point and two from another, and making equal angles with another plane perpendicular to these two, then if the first and third be parallel, the second and fourth will be likewise.

9. Draw a plane through a given straight line parallel to another given straight line.

10. Through a given point it is required to draw a planè parallel to both of two straight lines which do not intersect.

II.

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11. From a point above a plane two straight lines are drawn, the one at right angles to the plane, the other at right angles to a given line in that plane; shew that the straight line joining the feet of the perpendiculars is at right angles to the given line.

12. AB, AC, AD are three given straight lines at right angles to one another, AE is drawn perpendicular to CD, and BE is joined. Shew that BE is perpendicular to CD.

13. If perpendiculars AF, AF be drawn to a plane from two points A, A' above it, and a plane be drawn through A perpendicular to AA'; its line of intersection with the given plane is perpendicular to FF.

14. A, B, C, D are four points in space, AB, CD are at right angles to each other, and also AC, BD; shew that AD, BC will also be at right angles to one other.

15. Two planes intersect each other, and from any point in one of them a line is drawn perpendicular to the other, and also another line perpendicular to the line of intersection of both; shew that the plane which passes through these two lines is perpendicular to the line of intersection of the plane.

16. ABC is a triangle, the perpendiculars from A, B on the opposite sides meet in D, and through D is drawn a straight line perpendicular to the plane of the triangle; if E be any point in the line, shew that 、 EA, BC; EB, CA; and EC, AB are respectively perpendicular to

each other.

17. Find the distance of a given point from a given line in space. 18. Draw a line perpendicular to two lines which are not in the same plane.

19. Two planes being given perpendicular to each other, draw a third perpendicular to both.

III.

20. Two perpendiculars are let fall from any point on two given planes, shew that the angle between the perpendiculars will be equal to the angle of inclination of the planes to one another.

21. If through any point two straight lines be drawn equally inclined, the first to one plane and the second to another, shew that the angle between the lines is equal to the angle between the planes.

22. Two planes intersect, straight lines are drawn in one of the planes from a point in their common intersection making equal angles with it, shew that they are equally inclined to the other plane.

23. Two planes intersect at right angles in the line AB; at a point C in this plane are drawn CE and CF in one of the planes so that the angle ACE is equal to ACF. CE and CF will make equal angles with any line through C in the other plane.

IV.

24. Three straight lines not in the same plane, but parallel to and equidistant from each other, are intersected by a plane, and the points of intersection joined; shew when the triangle thus formed will be equilateral and when isosceles.

25. Three parallel straight lines are cut by parallel planes, and the points of intersection joined, the figures so formed are all similar and equal.

26. If a straight line PBpb cut two parallel planes in B, b, P and p being equidistant from the planes, and PAa, pcC be other lines drawn from P, p, to cut the planes, then the triangles ABC, abc will be equal to one another.

27. If two straight lines be cut by four parallel planes, the two segments, intercepted by the first and second planes, have the same ratio to each other as the two segments intercepted by the third and fourth planes.

28. ~ If three straight lines, which do not all lie in one plane, be cut in the same ratio by three planes, two of which are parallel, shew that the third will be parallel to the other two, if its intersections with the three straight lines are not all in one straight line.

29. To describe a circle which shall touch two given planes, and pass through a given point.

30. Three lines not in the same plane meet in a point; if a plane cut these lines at equal distances from the point of intersection, shew that the perpendicular from that point on the plane will meet it in the center of the circle inscribed in the triangle, formed by the portion of the plane intercepted by the planes passing through the lines.

31.

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A solid angle is contained by the planes BOC, COA, AOB: AD is drawn perpendicular to the plane BOC, and DB, DC are drawn in that plane perpendicular to OB, OC respectively: if AB, AC be joined, shew that they are perpendicular to OB, OC respectively.

32. Three straight lines, not in the same plane, intersect in a point, and through their point of intersection another straight line is drawn within the solid angle formed by them; prove that the angles which

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