For similar figures are to each other as the squares of their homologous sides; but the square of the side x is equal to the sum or the difference of the squares described upon the homologous sides A and B; therefore the figure described upon the side x is equivalent to the sum or to the difference of the similar figures described upon the sides A and B. PROBLEM XXXVII. 342. To construct a polygon similar to a given polygon, and which shall have to it a given ratio. Let A be a side of the given polygon. Find the side B of a square, which is to the square on A in the given ratio of the polygons (Prob. XXXIII.). Upon B construct a polygon similar to the given polygon (Prob. XXXV.), and B will be the polygon required. A For the similar polygons constructed upon A and B have the same ratio to each other as the squares constructed upon A and B (Prop. XXXI. Bk. IV.). PROBLEM XXXVIII. 343. To construct a polygon similar to a given polygon, P, and which shall be equivalent to another polygon, Q. Find M, the side of a square, equivalent to the polygon P, and N, the side of a square equivalent to the polygon Q. Let x be a fourth proportional to the three given lines A P B M, N, A B; upon the side x, homologous to A B, describe a polygon similar to the polygon P (Prob. XXXV.); it will also be equivalent to the polygon Q. For, representing the polygon described upon the side x by y, we have but, by construction, P: y :: A B2: x2; AB:x:: M: N, or A B2: x2:: M2 : N2; hence, Py:: M: No. But, by construction also, M2 is equivalent to P, and No is equivalent to Q; therefore, Py::P:Q; consequently y is equal to Q; hence the polygon y is similar to the polygon P, and equivalent to the polygon Q. BOOK VI. REGULAR POLYGONS, AND THE AREA OF THE CIRCLE. DEFINITIONS. 344. A REGULAR POLYGON is one which is both equilateral and equiangular. 345. Regular polygons may have any number of sides: the equilateral triangle is one of three sides; the square is one of four. PROPOSITION I.-THEOREM. 346. Regular polygons of the same number of sides are gons are similar. For, since the two polygons have the same number of sides, they have the same number of angles; and the sum of all the angles is the same in the one as in the other (Prop. XXIX. Bk. I.). Also, since the polygons are equiangular, each of the angles A, B, C, &c. is equal to each of the angles G, H, I, &c.; hence the two polygons are mutually equiangular. Again; the polygons being regular, the sides A B, BC, CD, &c. are equal to each other; so likewise are the sides GH, HI, IK, &c. Hence, AB: GH:: BC: HI:: CD: IK, &c. Therefore the two polygons have their angles equal, and their homologous sides proportional; hence they are similar (Art. 210). 347. Cor. The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop. XXXI. Bk. IV.). 348. Scholium. The angle of a regular polygon is determined by the number of its sides (Prop. XXIX. Bk. I.). PROPOSITION II.THEOREM. 349. A circle may be circumscribed about, and another inscribed in, any regular polygon. Let ABCDEFGH be any regular polygon; then a circle may be circumscribed about, and another inscribed in it. H G F A B C E D Describe a circle whose circumference shall pass through the three points A, B, C, the centre being 0; let fall the perpendicular OP from O to the middle point of the side BC; and draw the straight lines OA, OB, OC, OD. Now, if the quadrilateral OP CD be placed upon the quadrilateral OP BA, they will coincide; for the side OP is common, and the angle OPC is equal to the angle OPB, each being a right angle; consequently the side PC will fall upon its equal, P B, and the point C on B. Moreover, from the nature of the polygon, the angle PCD is equal to the angle PBA; therefore CD will take the direction B A, and CD being equal to BA, the point D will fall upon A, and the two quadrilaterals will coincide throughout. Therefore OD is equal to AO, and the circumference which passes through the three points A, B, C, will also pass through the point D. By the same mode of reasoning, it may be shown that the circle which passes through the three vertices B, C, D, will also pass through the vertex E, and so on. Hence, the circumference which passes through the three points A, B, C, passes through the vertices of all the angles of the polygon, and is circumscribed about the polygon (Art. 166). Again, with respect to this circumference, all the sides, AB, BC, CD, &c., of the polygon are equal chords; consequently they are equally distant from the centre (Prop. VIII. Bk. III.). Hence, if from the point O, as a centre, and with the radius OP, a circle be described, the circumference will touch the side B C, and all the other sides of the polygon, each at its middle point, and the circle will be inscribed in the polygon (Art. 168). 350. Scholium 1. The point O, the common centre of the circumscribed and inscribed circles, may also be regarded as the centre of the polygon. The angle formed at the centre by two radii drawn to the extremities of the same side is called the angle at the centre; and the perpendicular from the centre to a side is called the apothegm of the polygon. Since all the chords AB, BC, CD, &c. are equal, all the angles at the centre must likewise be equal; therefore the value of each may be found by dividing four right angles by the number of sides of the polygon. 351. Scholium 2. To inscribe a regular polygon of any number of sides in a given circle, it is only necessary to |