« ΠροηγούμενηΣυνέχεια »
AK being the diameter of the circle, AL.AK = BA, AD, and CL.AK = BC.CD; whence, by addition, AL. AK + CL.AK, or (II. 1.) AC.AK = BA.AD + BC.CD. In a similar manner, it would be proved, that BD.AK = AB.BC + AD.DC. Hence AC.AK: BD.AK, or (VI. 1.) AC: BD :: BA.AD + BC.CD: AB.BC + AD.DC.
But if AC be not perpendicular to BD, draw AEF perpendicular and CF parallel to BD, and DGH perpendicular and BH parallel to AC. Then, because EF is equal to the perpendicular drawn from C to BD, and GH equal to the one drawn from B to AC; it would be proved as before, that AF.AK = BA. AD + BC.CD, and DH.AK = AB.BC + AD. DC. Hence AF.AK : DHAK, or (VI. 1.) AF: DH:: BA.AD + BC.CD: AB.BC + AD.DC. But the triangles AFC, DHB are equiangular, having the right angles F and H, and the angles ACF, DBH, each equal (I. 29.) to ALD: therefore (VI. 4.) AF: AC :: DH: DB, and alternately AF: DH:: AC: DB. Hence (V. 11.) the foregoing analogy becomes AC: BD :: BA.AD + BC.CD : AB.BC + AD.DC. Wherefore, the diagonals, &c. *
PROP. G. THEOR. If in a straight line drawn through the centre of a circle, and on the same side of the centre, two points be taken so that the radius is a mean proportional between their distances from the centre; two straight lines drawn from those points to any point whatever in the circumference, are proportional to the segments into which the circumference divides the straight line intercepted between the same points.
Let ABC be a circle, and CAE a straight line drawn through
From this proposition and the last, when the sides of a quadrilateral inscribed in a circle are given, we can find the ratio of the diagonals and their rectangle, and thence (VI. 25.) the diagonals themselves. Also, if the sides be given in numbers, we can compute the diagonals. Thus, let the sides taken in succession round the figure be 50, 78, 104, and 120. Then, the ratio of the diagonals will be that of 50x78 + 104x120 to 50x120 + 78X104 ; that is, 16380 to 14112, or 65 to 56, by dividing by 252. Again, the rectangle of the diagonals is 50 X 104 +78X 120, or 14560. But (VÍ. 20. cor. 3.) similar rectilineal figures are as the squares of the corresponding sides, and consequently the sides are as the square roots of the areas : therefore, taking 65 and 56 as the sides of a rectangle, we have its area equal to 3640; and ✓3640 is to v 14560, or ✓3640 is to V(4X3640), that is, 1: 2: : 65 : 130 : : 56 : 112; so that 130 and 112 are the diagonals.
its centre D ; if ED: DA::DA: DF, and if BE, BF be drawn from any point B of the circumference; EB : BF:: EA: AF.
Join AB, BD. Then, since DB is equal to DA, we have (hyp.) ED : DB :: DB : DF. The two triangles EDB, BDF, therefore, have their sides about the common angle D proportional; wherefore (VI. 6.) the angle E is equal to FBD. Now the angle BAD is equal (I. 32.) to the two angles and EBA, and also (I. 5.) to ABD: wherefore E and EBA are (I. ax. 1.) equal to ABD. From these take the equal angles E and FBD, and (I. ax, 3.) the remaining angles EBA, ABF are equal : and therefore, (VI. 3.) in the triangle EBF, EB : BF :: EA: AF. If, therefore, in a straight line, &c.
Cor. Join BC, and produce EB to G. Then ABC being (III. 31.) a right angle, is equal to the two EBA, CBG. From these equals take the equal angles ABF, EBA, and the remainders FBC, CBG are equal : and therefore (VI. A.) EB : BF:: EC : CF. But it has been proved, that EB : BF :: EA : AF: therefore (V. 11.) EA : AF :: EC: CF. Hence, if the seg. ments EA, AF be given, the point C may be determined by the method shown in the scholium to the 10th proposition of this book; and the circle ABC may then be described, its diameter AC being determined. *
PROP. H. THEOR.
The perpendiculars drawn from the three angles of any triangle to the opposite sides, intersect one another in the same point.
Ifthe triangle be right-angled, it is plain that all the perpendiculars pass through the right angle. But if it be not right-angled, let ABC be the triangle, and about it describe a circle : then, B and C being acute angles, † draw ADE perpendicular to BC, cutting
• The circle may also be determined in the following manner.
Since (hyp.) ED: DA :: DA : DF, by division, EA : DA : : AF: DF; whence, alternately and by division, EA – AF: AF :: AF: DF. Hence DF is a third proportional to the difference of EA, AF, and to AF, the less ; and thus the centre D is determined. From the last analogy also we obtain (V. E. cor.) EA – AF: EA :: AF: AD; an analogy which serves the same purpose, since it shows that the radius of the circle is a fourth proportional to the difference of the segments EA, AF, and to thuse segments themselves.
+ This limitation prevents the necessity of a different case, which would arise if the perpendicular AD fell without the triangle. If the angle A be obtuse, the point F lies without the circle, and BF, not produced, cuts AC produced. The proof, however, is the same. The demonstration bere given is new, so far as is known to the present editor; and it is very easy and obvious. Another
BC in D, and the circumference in E; and make DF equal to DE: join BF and produce it, if necessary, to cut AC, or AC produced in G: BG is perpendicular to AC. Join BE: and because FD is equal to DE, the angles at D right angles, and DB common to the two triangles FDB, EDB, the angle FBD is equal (I. 4.) to EBD: but (III. 21.) CAD, EBD are also equal, because they are in the same segment: therefore CAD is equal to FBD or GBC.
But the angle ACB is common to the two triangles ACD, BCG ; and therefore (I. 32. cor. 5.) the remaining angles ADC, BGC are equal : but (const.) ADC is a right angle; therefore also BGC is a right angle, and BG is perpendicular to AC. In the same manner it would be shown, that a straight line CH, drawn through C and F, is perpendicular to AB. The three perpendiculars therefore all pass through F : wherefore the perpendiculars, &c.
easy and elegant proof, of which the following is an outline, is given in Garnier's “ Réciproques," &c., theor. III. page 78: Draw BG and CH perpendicular to AC and AB;' join GH, and about the quadrilaterals AHFG and BHGC describe circles, which can be done, as is easily shown : draw also AFD. Then the angles BAD, BCH are equal, each of them being equal (III. 21.) to HGF: and the angle ABC being common, ADB is equal (I. 32.) to BHC, and is therefore a right angle.
1. A STRAIGHT line is said to be perpendicular to a plane when it makes right angles with all straight lines meeting it in that plane.
2. The inclination of two planes which meet one another, is the angle contained by two straight lines drawn from any point of their common section at right angles to it, one upon each plane.
3. If that angle be a right angle, the planes are perpendicular to one another.
4. Parallel planes are such as do not meet one another, though produced ever so far in every direction.
5. A solid angle is that which is made by more than two plane angles meeting in one point, and not lying in the same plane.
If the number of plane angles be three, the solid angle is trihedral ; if four, tetrahedral ; if more than four, polyhedral.
6. A polyhedron is a solid figure contained by plane figures.
If it be contained by four plane figures, it is called a tetrahedron ; if by six, a hexahedron ; if by eight, an octahedron ; if by twelve, a dodecahedron ; if by twenty an icosahedron, &c.
7. A regular body, or regular polyhedron, is a solid contained by plane figures, which are ah equal and similar.
8. Of solid figures contained by planes, those are similar, which have all their solid angles equal, each to each, and which are contained by the same number of similar plane figures, similarly situated.
9. A pyramid is a solid figure contained by one plane figure, called its base, and by three or more triangles meeting in a point without the plane, called the vertex of the pyramid.
10. A prism is a solid figure, the ends or bases of which are parallel, and are equal and similar plane figures, and its other boundaries are parallelograms. One of these parallelograms also is sometimes regarded as the base of the prism.
• The angle which one plane makes with another, is sometimes called a dihedral angle.
11. Pyramids and prisms are said to be triangular, when their bases are triangles ; quadrangular, when their bases are quadrilaterals; pentagonal, when they are pentagons, &c.
12. The altitude of a pyramid is the perpendicular drawn from its vertex to its base: and the altitude of a prism is either a perpendicular drawn from any point in one of its ends or bases to the other; or a perpendicular to one of its bounding parallelograms from a point in the line opposite. The first of these altitudes is sometimes called the length of the prism.
13. A prism, of which the ends or bases are perpendicular to the other sides, is called a right prism: any other is an oblique prism.
14. A parallelepiped is a prism of which the bases are parallelograms.
15. A parallelepiped of which the bases and the other sides are rectangles, is said to be rectangular.
16. A cube is a rectangular parallelepiped, which has all its six sides squares.
17. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.
18. The axis of a sphere is the fixed straight line about which the semicircle revolves.
19. The centre of a sphere is the same as that of the generating semicircle.
20. A diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by its surface.
21. A cone is a solid figure described by the revolution of a right-angled triangle about one of the legs, which remains fixed.
If the fixed leg be equal to the other leg, the cone is called a right-angled cone; if it be less than the other leg, an obtuse-angled, and if greater, an acute-angled cone.
22. The axis of a cone is the fixed straight line about which the triangle revolves.
23. The base of a cone is the circle described by the leg which revolves.
24. A cylinder is a solid figure described by the revolution of a rectangle about one of its sides, which remains fixed.
25. The axis of a cylinder is the fixed straight line about which the rectangle revolves.
26. The bases, or ends, of a cylinder are the circles described by the two revolving opposite sides of the rectangle.
27. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.
PROP. I. THEOR. One part of a straight line cannot be in a plane, and another part above it.
If it be possible, let AB, a part of the straight line ABC, be in the plane, and the part BC above it: and (I. post. 2.) since AB