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37. The vertical angle C of a triangle is bisected by a straight line which meets the base at D, and is produced to a point E, such that the rectangle contained by CD and CE is equal to the rectangle contained by AC and CB: shew that if the base and vertical angle be given, the position of E is invariable.

38.

ABC is an isosceles triangle having the base angles at B and C each double of the vertical angle: if BE and CD bisect the base angles and meet the opposite sides in E and D, shew that DE divides the triangle into figures whose ratio is equal to that of AB to BC.

39. If AB, the diameter of a semicircle, be bisected in C and on AC and CB circles be described, and in the space between the three circumferences a circle be inscribed, shew that its diameter will be to that of the equal circles in the ratio of two to three.

40. O is the centre of a circle inscribed in a quadrilateral ABCD; a line EOF is drawn and making equal angles with AD and BC, and meeting them in E and F respectively: shew that the triangles AEO, BOF are similar, and that

AE: ED=CF : FB.

41. From the last exercise deduce the following: The inscribed circle of a triangle ABC touches AB in F; XOY is drawn through the centre making equal angles with AB and AC, and meeting them in X and Y respectively: shew that BX: XF = AY : YC.

42. Inscribe a square in a given semicircle.

43. Inscribe a square in a given segment of a circle.

44. Describe an equilateral triangle equal to a given isosceles triangle.

45. Describe a square having given the difference between a diagonal and a side.

46. Given the vertical angle, the ratio of the sides containing it, and the diameter of the circumscribing circle, construct the triangle.

47. Given the vertical angle, the line bisecting the base, and the angle the bisector makes with the base, construct the triangle.

48. In a given circle inscribe a triangle so that two sides may pass through two given points and the third side be parallel to a given straight line.

49. In a given circle inscribe a triangle so that the sides may pass through three given points.

50. A, B, X, Y are four points in a straight line, and O is such a point in it that the rectangle OA, OY is equal to the rectangle OB, OX: if a circle be described with centre O and radius equal to a mean proportional between OA and OY, shew that at every point on this circle AB and XY will subtend equal angles.

51. O is a fixed point, and OP is any line drawn to meet a fixed straight line in P; if on OP a point Q is taken so that OQ to OP is a constant ratio, find the locus of Q.

52. O is a fixed point, and OP is any line drawn to meet the circumference of a fixed circle in P; if on OP a point Q is taken so that OQ to OP is a constant ratio, find the locus of Q.

53. If from a given point two straight lines are drawn including a given angle, and having a fixed ratio, find the locus of the extremity of one of them when the extremity of the other lies on a fixed straight line.

54. On a straight line PAB, two points A and B are marked and the line PAB is made to revolve round the fixed extremity P. C is a fixed point in the plane in which PAB revolves; prove that if CA and CB be joined and the parallelogram CADB be completed, the locus of D will be a circle.

55. Find the locus of a point whose distances from two fixed points are in a given ratio.

56. Find the locus of a point from which two given circles subtend the same angle.

57. Find the locus of a point such that its distances from two intersecting straight lines are in a given ratio.

58. In the figure on page 364, shew that QT, P'T' meet on the radical axis of the two circles.

59. ABC is any triangle, and on its sides equilateral triangles are described externally if X, Y, Z are the centres of their inscribed circles, shew that the triangle XYZ is equilateral.

60. If S, I are the centres, and R, r the radii of the circumscribed and inscribed circles of a triangle, and if N is the centre of its ninepoints circle,

prove that (i) SI2=R2-2Rr,

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Establish corresponding properties for the escribed circles, and hence prove that the nine-points circle touches the inscribed and escribed ciroles of a triangle.

SOLID GEOMETRY.

EUCLID. BOOK XI.

DEFINITIONS.

FROM the Definitions of Book I. it will be remembered that

(i) A line is that which has length, without breadth or thickness.

(ii) A surface is that which has length and breadth, without thickness.

To these definitions we have now to add :

(iii) Space is that which has length, breadth, and thickness.

Thus a line is said to be of one dimension;

a surface is said to be of two dimensions;
and space is said to be of three dimensions.

The Propositions of Euclid's Eleventh Book here given establish the first principles of the geometry of space, or solid geometry. They deal with the properties of straight lines which are not all in the same plane, the relations which straight lines bear to planes which do not contain those lines, and the relations which two or more planes bear to one another. Unless the contrary is stated the straight lines are supposed to be of indefinite length, and the planes of infinite extent.

Solid geometry then proceeds to discuss the properties of solid figures, of surfaces which are not planes, and of lines which can not be drawn on a plane surface.

LINES AND PLANES.

1. A straight line is perpendicular to a plane when it is perpendicular to every straight line which meets it in that plane.

[graphic]

NOTE. It will be proved in Proposition 4 that if a straight line is perpendicular to two straight lines which meet it in a plane, it is also perpendicular to every straight line which meets it in that plane.

A straight line drawn perpendicular to a plane is said to be a normal to that plane.

2. The foot of the perpendicular let fall from a given point on a plane is called the projection of that point on the plane.

3.

The projection of a line on a plane is the locus of the feet of perpendiculars drawn from all points in the given line to the plane.

[graphic][subsumed]

Thus in the above figure the line ab is the projection of the line AB on the plane PQ.

It will be proved hereafter (see page 420) that the projection of a straight line on a plane is also a straight line.

4. The inclination of a straight line to a plane is the acute angle contained by that line and another drawn from the point at which the first line meets the plane to the point at which a perpendicular to the plane let fall from any point of the first line meets the plane.

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Thus in the above figure, if from any point X in the given straight line AB, which intersects the plane PQ at A, a perpendicular Xx is let fall on the plane, and the straight line Axb is drawn from A through x, then the inclination of the straight line AB to the plane PQ is measured by the acute angle BAb. In other words :

The inclination of a straight line to a plane is the acute angle contained by the given straight line and its projection on the plane.

AXIOM. If two surfaces intersect one another, they meet in a line or lines.

5. The common section of two intersecting surfaces is the line (or lines) in which they meet.

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NOTE. It is proved in Proposition 3 that the common section of two planes is a straight line.

Thus AB, the common section of the two planes PQ, XY is proved to be a straight line.

H. E.

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