B Let ABCDE be the given polygon, and FG the given straight line. Draw the diagonals AC, AD. At the point F in E K the straight line F G, make the angle GFH equal to the angle BAC; and at the point G make the angle F G H equal to the angle ABC. The lines FH, GH will cut each other in H, and FGH will be a triangle similar to ABC. In the same manner, upon FH, homologous to A C, construct the triangle FIH similar to A DC; and upon FI, homologous to AD, construct the triangle FIK similar to A D E. The polygon FGHIK will be similar to ABCDE, as required. A 4 C Let A and B be two homologous sides of the given poly D G F For these two polygons are composed of the same number of triangles, similar each to each, and similarly situated (Prop. XXX. Cor., Bk. IV.). H PROBLEM XXXVI. 341. Two similar polygons being given, to construct a similar polygon, which shall be equivalent to their sum or their difference. B gons. Find a square equal to the sum or to the difference of the squares described up A on A and B; let x be the side of that square; then will x in the polygon required be the side which is homologous to the sides A and B in the given polygons. The polygon itself may then be constructed on x, by the last problem. 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 B M, N, AB; 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. P ୪ but, by construction, For, representing the polygon described upon the side x by y, we have Py:: AB2: 22; hence, AB:x:: M: N, or AB2: 22:: M2: N2; Py:: M2: No. But, by construction also, M2 is equivalent to P, and N2 is equivalent to Q; therefore, P: y::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 similar figures. Let ABCDEF, GHIKLM, be two regular polygons of the same number of sides; then these polygons are similar. A B G H 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. F E D C M L K 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 A F E 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. B C 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 OP B, each being a right angle; consequently the side PC will fall upon its equal, PB, and the point C on B. Morcover, from the nature of the polygon, the angle PCD is equal to the angle PBA; therefore CD will take the D |