From C let fall the I CK; and from D as a centre, with a radius equal to D B, describe an arc cutting H B produced, at P. verimeter § 113 Now BK = } BH, PROPOSITION XXVII. THEOREM. 410. The maximum of isoperimetrical polygons of the same number of sides is equilateral. B K Let A B CD, etc., be the maximum of isoperimetrical polygons of any given number of sides. Draw A C. The A ABC must be the maximum of all the A which are formed upon A C with a perimeter equal to that of A A BC. Otherwise, a greater A A KC could be substituted for AA BC, without changing the perimeter of the polygon. But this is inconsistent with the hypothesis that the polygon A B C D, etc., is the maximum polygon. .:: the A ABC, is isosceles, § 409 (of all s having the same base and equal perimeters, the isosceles A is the maximum). Q. E. D. 411. COROLLARY. The maximum of isoperimetrical polygons of the same number of sides is a regular polygon. For, it is equilateral, § 410 (the maximum of isoperimetrical polygons of the same number of sides is equilateral). Also it can be inscribed in a 0, $ 408 (the maximum of all polygons formed of given sides can be inscribed in a O). Hence it is regular, § 364 (an equilateral polygon inscribed in a О is regular). PROPOSITION XXVIII. THEOREM. 412. Of isoperimetrical regular polygons, that is greatest which has the greatest number of sides. B Let Q be a regular polygon of three sides, and Q' be a regular polygon of four sides, each having the same perimeter. The polygon may be considered an irregular polygon of four sides, in which the sides A D and D B make with each other an 2 equal to two rt. 6. Then the irregular polygon Q, of four sides is less than the regular isoperimetrical polygon Q' of four sides, § 411 (the maximum of isoperimetrical polygons of the same number of sides is a regular polygon). In like manner it may be shown that Q is less than a regular isoperimetrical polygon of five sides, and so on. Q. E. D. 413. COROLLARY. Of all isoperimetrical plane figures the circle is the maximum. PROPOSITION XXIX. THEOREM. 414. If a regular polygon be constructed with a given area, its perimeter will be the less the greater the number of its sides. Let Q and Q be regular polygons having the same area, and let Q' have the greater number of sides. We are to prove the perimeter of Q > the perimeter of Q'. Let Q" be a regular polygon having the same perimeter as Q', and the same number of sides as Q. Then Q is > Q", § 412 (of isoperimetrical regular polygons, that is the greatest which has the greatest number of sides). But Q=Q, .: Q is > Q". ::: the perimeter of Q is > the perimeter of Q". But the perimeter of Q=the perimeter of Q", Cons. .. the perimeter of Q is > that of Q'. Q. E. D. 415. COROLLARY. The circumference of a circle is less than the perimeter of any other plane figure of equal area. ON SYMMETRY. — SUPPLEMENTARY. 416. Two points are Symmetrical when they are situated on opposite sides of, and at equal distances from, a fixed point, line, or plane, taken as an object of reference. 417. When a point is taken as an object of reference, it is called the Centre of Symmetry; when a line is taken, it is called the Axis of Symmetry; when a plane is taken, it is called the Plane of Symmetry. 418. Two points are symmetrical with re. spect to a centre, if the centre bisect the straight line terminated by these points. Thus, P, P' are symmetrical with respect to C, if C bisect the straight line P P. PI 419. The distance of either of the two synimetrical points from the centre of symmetry is called the Radius of Symmetry. Thus either C P or C P' is the radius of symmetry. X 420. Two points are symmetrical with respect to an axis, if the axis bisect at right angles the straight line terminated by these X-F points. Thus, P, P are symmetrical with respect to the axis X X', if X X' bisect P P' at right angles. 421. Two points are symmetrical with respect to a plane, if the plane bisect at right angles the straight line terminated by these points. Thus P, P' are symmetrical with respect to M N, if M N bisect P P at M right angles. |