488. Of isoperimetric polygons of the same number of sides, the maximum is equilateral. Let ABCD etc. be the maximum of isoperimetric polygons of any given number of sides. To prove that AB, BC, CD, etc., are equal. The ▲ ABC must be the maximum of all the ▲ which are formed upon AC with a perimeter equal to that of ▲ ABC. Otherwise a greater ▲ AKC could be substituted for ▲ ABC, without changing the perimeter of the polygon. But this is inconsistent with the hypothesis that the polyon ABCD etc. is the maximum polygon. .. the ▲ ABC is isosceles. .. AB = BC. In like manner it may be proved that BC=CD, etc. § 485 Q. E. D. 489. COR. The maximum of isoperimetric polygons of the same number of sides is a regular polygon. For the maximum polygon is equilateral (§ 488), and can be inscribed in a circle (§ 487), and is, therefore, regular. § 430 Q. E. D. PROPOSITION XXV. THEOREM. 490. Of isoperimetric regular polygons, that which has the greatest number of sides is the maximum. Let Q be a regular polygon of three sides, and Q' a regular polygon of four sides, and let the two polygons have equal perimeters. To prove that Q' is greater than Q. Proof. Draw CD from C to any point in AB. Invert the ▲ CDA and place it in the position DCE, letting D fall at C, C at D, and A at E. The polygon DBCE is an irregular polygon of four sides, which by construction has the same perimeter as Q', and the same area as Q. Then the irregular polygon DBCE of four sides is less than the isoperimetric regular polygon Q' of four sides. § 489 In like manner it may be shown that Q' is less than an isoperimetric regular polygon of five sides, and so on. Q. E.D. Ex. 445. Of all equivalent parallelograms that have equal bases, the rectangle has the minimum perimeter. Ex. 446. Of all equivalent rectangles, the square has the minimum perimeter. Ex. 447. Of all triangles that have the same base and the same altitude, the isosceles has the minimum perimeter. Ex. 448. Of all triangles that can be inscribed in a given circle, the equilateral is the maximum and has the maximum perimeter. PROPOSITION XXVI. THEOREM. 491. Of regular polygons having a given area, that which has the greatest number of sides has the least perimeter. Let Q and Q' be regular polygons having the same area, and let Q' have the greater number of sides. To prove the perimeter of Q> the perimeter of Q'. Proof. Let Q" be a regular polygon having the same perimeter as Q', and the same number of sides as Q. Then Q' > Q", § 490 (of isoperimetric regular polygons, that which has the greatest number of sides is the maximum). But Q ~ Q'. .. Q > Q". Hyp. .. the perimeter of Q> the perimeter of Q". But the perimeter of Q' the perimeter of Q". Hyp. .. the perimeter of Q> the perimeter of Q'. Q. E. D. Ex. 449. To inscribe in a semicircle the maximum rectangle. Ex. 450. Of all polygons of a given number of sides which may be inscribed in a given circle, that which is regular has the maximum area and the maximum perimeter. Ex. 451. Of all polygons of a given number of sides which may be circumscribed about a given circle, that which is regular has the minimum area and the minimum perimeter. THEOREMS. Ex. 452. Every equilateral polygon circumscribed about a circle is regular if it has an odd number of sides. Ex. 453. Every equiangular polygon inscribed in a circle is regular if it has an odd number of sides. Ex. 454. Every equiangular polygon circumscribed about a circle is regular. Ex. 455. The side of a circumscribed equilateral triangle is equal to twice the side of the similar inscribed triangle. Ex. 456. The apothem of an inscribed regular hexagon is equal to half the side of the inscribed equilateral triangle. Ex. 457. The area of an inscribed regular hexagon is three fourths of the area of the circumscribed regular hexagon. Ex. 458. The area of an inscribed regular hexagon is the mean proportional between the areas of the inscribed and the circumscribed equilateral triangles. Ex. 459. The square of the side of an inscribed equilateral triangle is equal to three times the square of a side of the inscribed regular hexagon. Ex. 460. The area of an inscribed equilateral triangle is equal to half the area of the inscribed regular hexagon. Ex. 461. The square of the side of an inscribed equilateral triangle is equal to the sum of the squares of the sides of the inscribed square and of the inscribed regular hexagon. Ex. 462. The square of the side of an inscribed regular pentagon is equal to the sum of the squares of the radius of the circle and the side of the inscribed regular decagon. If R denotes the radius of a circle, and a one side of an inscribed regular polygon, show that: Ex. 463. In a regular pentagon, a = R√10 – 2 √5. Ex. 464. In a regular octagon, a = R √2 – √2. Ex. 465. In a regular dodecagon, a = R √2 – √3. Ex. 466. If two diagonals of a regular pentagon intersect, the longer segment of each is equal to a side of the pentagon. Ex. 467. The apothem of an inscribed regular pentagon is equal to half the sum of the radius of the circle and the side of the inscribed regular decagon. Ex. 468. The side of an inscribed regular pentagon is equal to the hypotenuse of the right triangle which has for legs the radius of the circle and the side of the inscribed regular decagon. Ex. 469. The radius of an inscribed regular polygon is the mean proportional between its apothem and the radius of the similar circumscribed regular polygon. Ex. 470. If squares are constructed outwardly upon the six sides of a regular hexagon, the exterior vertices of these squares are the vertices of a regular dodecagon. Ex. 471. If the alternate vertices of a regular hexagon are joined by straight lines, show that another regular hexagon is thereby formed. Find the ratio of the areas of these two hexagons. Ex. 472. If on the legs of a right triangle as diameters semicircles are described external to the triangle, and from the whole figure a semicircle on the hypotenuse is subtracted, the remaining figure is equivalent to the given right triangle. Ex. 473. The star-shaped polygon, formed by producing the sides of a regular hexagon, is equivalent to twice the given hexagon. Ex. 474. The sum of the perpendiculars drawn to the sides of a regular polygon from any point within the polygon is equal to the apothem multiplied by the number of sides. Ex. 475. If two chords of a circle are perpendicular to each other, the sum of the four circles described on the four segments as diameters is equivalent to the given circle. Ex. 476. If the diameter of a circle is divided into any two segments, and upon these segments as diameters semicircumferences are described upon opposite sides of the diameter, these semicircumferences divide the circle into two parts which have the same ratio as the two segments of the diameter. Ex. 477. The diagonals that join any vertex of a regular polygon to all the vertices not adjacent divide the angle at that vertex into as many equal parts less two as the polygon has sides. |