380. Cor. Of isoperimetric triangles, that which is equi lateral is the maximum. For if the maximum ▲ is not isosceles when any side is taken as the base, its area can be increased by making it isosceles. Then, the maximum ▲ is equilateral. PROP. III. THEOREM. (§ 379) 381. Of isoperimetric polygons having the same number of sides, that which is equilateral is the maximum. Given ABCDE the maximum of polygons having the given perimeter and the given number of sides. To Prove ABCDE equilateral. Proof. If possible, let sides AB and BC be unequal. Let AB'C be an isosceles ▲ with the base AC, having its perimeter equal to that of ▲ ABC. .. area AB'C> area ABC. Adding area ACDE to both members, area AB'CDE > area ABCDE. (§ 379) But this is impossible; for, by hyp., ABCDE is the maximum of polygons having the given perimeter. Hence, AB and BC cannot be unequal. In like manner we have BCCD= DE, etc. Then, ABCDE is equilateral. PROP. IV. THEOREM. 382. Of isoperimetric equilateral polygons having the same number of sides, that which is equiangular is the maximum. Given AB, BC, and CD any three consecutive sides of the maximum of isoperimetric equilateral polygons having the same number of sides. If possible, let ≤ ABC be >< BCD, and draw line AD. In Fig. 1. Let E be the middle point of BC; and draw line EF, meeting AB produced at F, making EF = BE. Produce FE to meet CD at G. Then in ABEF and CEG, by hyp., BE CE. Also, L BEF=Z CEG. (?) Lay off, on BH, FH = CH; and on DH, GH = BH; and draw line FG cutting BC at E. (§ 99) since BF = BH – FH, and CG =GH-CH. Lay off, on KB produced, FK=CK; and on CK, GK=BK; and draw line FG cutting BC at E. Then, in ▲ BEF and CEG, L F = Z C. (?) (?) (?) (?) (?) = Then since, in either figure, BC + CG BF + FG, and A BEFACEG, quadrilateral AFGD is isoperimetric with, and to, quadrilateral ABCD. Calling the remainder of the given polygon P, it follows that the polygon composed of AFGD and P is isoperimetric with, and to, the polygon composed of ABCD and P; that is, the given polygon. Then the polygon composed of AFGD and P must be the maximum of polygons having the given perimeter and the given number of sides. Hence, the polygon composed of AFGD and P is equilateral. But this is impossible, since AF is > DG. In like manner, Z ABC cannot be << BCD. .. LABC = ≤ BCD. (§ 381) Note. The case of triangles was considered in § 380. Fig. 3 also provides for the case of triangles by supposing D and K to coincide with A. In the case of quadrilaterals, P = 0. 383. Cor. Of isoperimetric polygons having the same number of sides, that which is regular is the maximum. PROP. V. THEOREM. 384. Of two isoperimetric regular polygons, that which has the greater number of sides has the greater area. Given ABC an equilateral A, and M an isoperimetric square. To Prove area Marea ABC. Proof. Let D be any point in side AB of ▲ ABC. Draw line DC; and construct isosceles ACDE isoperi metric with ▲ BCD, CD being its base. .. area CDE > area BCD. .. area ADEC > area ABC. But, since ADEC and M are isoperimetric, area M > area ADEC. .. area M> area ABC. (§ 379) (§ 381) In like manner, we may prove the area of a regular pentagon greater than that of an isoperimetric square; etc. 385. Cor. The area of a circle is greater than the area of any polygon having an equal perimeter. SYMMETRICAL FIGURES. DEFINITIONS. 386. Two points are said to be symmetrical with respect to a third, called the centre of symmetry, when the latter bisects the straight line which joins them. Thus, if O is the middle point of straight line AB, points A and B are symmetrical with respect to O A 0 as a centre. B 387. Two points are said to be symmetrical with respect to a straight line, called the axis of symmetry, when the latter bisects at right angles the straight line which joins them. Thus, if line CD bisects line AB at right angles, points A and B are symmetrical with respect to CD as an axis. G -D 388. Two figures are said to be symmetrical with respect to a centre, or with respect to an axis, when to every point of one there corresponds a symmetrical point in the other. |