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 389, 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 nine-points circle, (i) SIP=R2 – 2Rr, (ii) NI = {R - r. Establish corresponding properties for the escribed circles, and hence prove that the nine-points circle touches the inscribed and escribed circles of a triangle. prove that 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 cannot be drawn on a plane surface. LINES AND PLANES. A straight line is perpendicular to a plane when it is perpendicular to every straight line which meets it in that plane. 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. B А, P Thus in the above figure the line ab is the projection of the line AB on the plane PQ. NOTE. It will be proved hereafter (see page 446) 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. B A P 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. 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. 6. One plane is perpendicular to another plane when any straight line drawn in one of the planes perpendicular to the common section is also perpendicular to the other plane. Thus in the above figure, the plane EB is perpendicular to the plane CD, if any straight line PQ, drawn in the plane EB at right angles to the common section AB, is also at right angles to the plane CD. 7. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any point in the common section at right angles to it, one in one plane and one in the other. Thus in the adjoining figure, the straight line AB is the common section of the two intersecting planes BC, AD; and from Q, any point in AB, two straight lines QP, QR are drawn perpendicular to AB, one in each plane : then the inclination of the two planes is measured by the acute angle PQR. A B D NOTE. This definition assumes that the angle PQR is of constant magnitude whatever point Q is taken in AB: the truth of which assumption is proved in Proposition 10. The angle formed by the intersection of two planes is called a dihedral angle. It may be proved that two planes are perpendicular to one another when the dihedral angle formed by them is a right angle. |