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59. Of all isoperimetric polygons having the same number of sides, the regular polygon is the maximum.
1st. The maximum polygon P, of all the isoperimetric polygons of the same number of sides must have its sides equal; for if two of its sides, as AB', B'C, were unequal, we could, by (53), substitute for the triangle AB'C the isosceles triangle ABC having the same perimeter as AB'C' and a greater area, and thus the area of the whole polygon could be increased with out changing the length of its perimeter or the number of its sides.
2d. The maximum polygon constructed with the same number of equal sides must, by (58), be inscriptible in a circle; therefore it must be a regular polygon.
60. Of all polygons having the same number of sides and the same area, the regular polygon has the minimum perimeter.
Let P be a regular polygon, and M any irregular polygon having the same number of sides and the same area as P; then, the perimeter of P is less than that of M.
For, let N be a regular polygon having the same perimeter and the same number of sides as M; then, by (59), M < N, or P < N. But of two regular polygons having the same number of sides, that which has the less area has the less perimeter; therefore the perimeter of P is less than that of N, or less than that of M.
61. If a regular polygon be constructed with a given perimeter, its area will be the greater, the greater the number of its sides.
Let P be the regular polygon of three sides, and Q the regular polygon of four sides, constructed with the same given perimeter. In any side AB of P take any arbitrary point D; the polygon P may be regarded as an irregular polygon of four sides, in which the sides AD, DB, make an angle with each other equal to two right angles (I. 16); then, the irregular polygon P of four sides is less than the regular isoperimetric polygon Q of four sides (59). In the same manner it follows that Q is less than the regular isoperimetric polygon of five sides, and so on.
62. If a regular polygon be constructed with a given area, its peri'neter will be the less, the greater the number of its sides.
Let P and Q be regular polygons having the same area, and let Q have the greater number of sides; then, the perimeter of P will be greater than that
For, let R be a regular polygon having the same perimeter as Q and the same number of sides as P; then, by (61), Q > R, or P > R; therefore the perimeter of P is greater than that of R, or greater than that of Q.
GEOMETRY OF SPACE.
THE PLANE POLYEDRAL ANGLES.
1. Definition. A plane has already been defined as a surface such that the straight line joining any two points in it lies wholly in the surface.
Thus, the surface MN is a plane, if, A and B being any two points in it, the straight line AB lies wholly in the surface.
The plane is understood to be indefinite in extent, so that, however far the straight line is produced, all its points lie in the plane. But to represent a plane in a diagram, we are obliged to take a limited portion of it, and we usually represent it by a parallelogram supposed to lie in the plane.
DETERMINATION OF A PLANE.
2. Through any given straight line an infinite number of planes may be passed.
Let AB be a given straight line. A straight line may be drawn in any plane, and the position of that plane may be changed until the line drawn in it is brought into coincidence with AB. We shall then have one plane
passed through AB; and this plane may be turned upon AB as an axis and made to occupy an infinite number of positions.
3. Scholium. Hence, a plane subjected to the single condition that it shall pass through a given straight line, is not fixed, or determinate, in position. But it will become determinate if it is required to pass through an additional point, or line, as shown in the next proposition.
A plane is said to be determined by given lines, or points, when it is the only plane which contains such lines or points.
4. A plane is determined, 1st, by a straight line and a point without that line; 2d, by two intersecting straight lines; 3d, by three points not in the same straight line ; 4th, by two parallel straight lines.
1st. A plane MN being passed through a given straight line AB, and then turned upon this line as an axis until it contains a given point C not in the line AB, is evidently determined; for, if it is then turned in either direction about AB, it will cease to contain the point C. The plane is therefore determined by the given straight line and the point without it.
2d. If two intersecting straight lines AB, AC, are given, a plane passed through AB and any point C (other than the point A) of AC, contains the two straight lines, and is determined by these lines.
3d. If three points are given, A, B, C, not in the same straight line, any two of them may be joined by a straight line, and then the plane passed through this line and the third point, contains the three points, and is thus determined by them.
4th. Two parallel lines, AB, CD, are by definition (I. 42) necessarily in the same plane, and there is but one plane containing them, since a plane passed through one of them, AB, and any point E of the other, is determined in position.
5. Corollary. The intersection of two planes is a straight line. For, the intersection cannot contain three points not in the same • straight line, sii ce only one plane can contain three such points.
PERPENDICULARS AND OBLIQUE LINES TO PLANES.
6. Definition. A straight line is perpendicular to a plane when it is perpendicular to every straight line drawn in the plane through its foot, that is, through the point in which it meets the plane.
In the same case, the plane is said to be perpendicular to the line.
7. From a given point without a plane, one perpendicular to the plane can be drawn, and but one.
Let A be the given point, and MN the plane.
If any straight line, as AB, is drawn from A to a point B of the plane, and the point B is then supposed to move in the plane, the length of AB will vary. Thus, if B move along a straight line BB' in the plane, the distance AB will vary according to the distance of B from the foot of the perpendicular AC let fall from A upon BB'. Now, of all the lines drawn from A to points in the plane, there must be one minimum, or shortest line. There cannot be two equal shortest lines; for if AB and AB' are two equal straight lines from A to the plane, each is greater than the perpendicular AC let fall from A upon BB'; hence they are not minimum lines. There is therefore one, and but one, minimum line from A to the plane. Let AP be that minimum line; then, AP is perpendicular to any straight line EF drawn in the plane through its foot P. For, in the plane of the lines AP and EF, AP is the shortest line that can be drawn from A to any point in EF, since it is the shortest line that can be drawn from A to any point in the plane MN; therefore, AP is perpendicular to EF (I. 28). Thus AP is perpendicular to any, that is, to every, straight line drawn in the plane through its foot, and is therefore perpendicular to the plane. Moreover, by the nature of the proof just given, AP is the only perpendicular that can be drawn from A to the plane MN.
8. Corollary. At a given point Pin a plane MN, a perpendicular can be erected to the plane, and but one.