PROPOSITION XXII. PROBLEM. 352. To construct a square which shall have a given ratio to a given square. S It is required to construct a square which shall be to R as n is to m. = On a straight line take A B m, and BC= n. On A C as a diameter, describe a semicircle. Then the AA SC is a rt. A with the rt. at S, § 204 On SA, or SA produced, take SE equal to a side of R. (a straight line drawn through two sides of a ▲, parallel to the third side, divides those sides proportionally). that is, the square having a side equal to SF will have the same ratio to the square R, as n has to m. Q. E. F. 353. To construct a polygon similar to a given polygon and having a given ratio to it. It is required to construct a polygon similar to R, which shall be to R as n is to m. Find a line, A' B', such that the square constructed upon it shall be to the square constructed upon A B as n is to m. § 352 Upon A'B' as a side homologous to A B, construct the polygon S similar to R. (similar polygons are to each other as the squares on their homologous sides). PROPOSITION XXIV. PROBLEM. 354. To construct a square equivalent to a given paral Let ABCD be a parallelogram, b its base, and a its altitude. It is required to construct a square = ABCD. Upon the line MX take MN = a, and NO = b. Upon MO as a diameter, describe a semicircle. - At N erect NPL to MO. Then the square R, constructed upon a line equal to NP, is equivalent to the ABC D. For MN: NP :: NP: NO, $307 (alet fall from any point of a circumference to the diameter is a mean proportional between the segments of the diameter). (the product of the means is equal to the product of the extremes). Q. E. F. 355. COROLLARY 1. A square may be constructed equivalent to a triangle, by taking for its side a mean proportional between the base and one-half the altitude of the triangle. 356. COR. 2. A square may be constructed equivalent to any polygon, by first reducing the polygon to an equivalent triangle, and then constructing a square equivalent to the triangle. PROPOSITION II. THEOREM. 365. I. A circle may be circumscribed about a regular polygon. II. A circle may be inscribed in a regular polygon. Let ABCD, etc., be a regular polygon. L We are to prove that a O may be circumscribed about this regular polygon, and also a ○ may be inscribed in this regular polygon. CASE I. Describe a circumference passing through A, B, and C. From the centre 0, draw O A, O D, and draw Os 1 to chord BC. On Os as an axis revolve the quadrilateral O A Bs, until it comes into the plane of Os CD. The line s B will fall upon s C, (for LOs B = 40s C, both being rt. 4). The point B will fall upon C, The line BA will fall upon CD, § 183 (since LB = L C, being of a regular polygon). § 363 $363 The point A will fall upon D, = (since B A C D, being sides of a regular polygon). .. the line OA will coincide with line O D, .. the circumference will pass through D. In like manner we may prove that the circumference, passing through vertices B, C, and D will also pass through the vertex E, and thus through all the vertices of the polygon in succession. CASE II. The sides of the regular polygon, being equal chords of the circumscribed O, are equally distant from the centre, § 185 .. a circle described with the centre 0 and a radius Os will touch all the sides, and be inscribed in the polygon. § 174 366. DEF. The Centre of a regular polygon is the common centre of the circumscribed and inscribed circles. . 367. DEF. The Radius of a regular polygon is the radius OA of the circumscribed circle. 368. DEF. The Apothegm of a regular polygon is the radius Os of the inscribed circle. 369. DEF. The Angle at the centre is the angle included by the radii drawn to the extremities of any side. PROPOSITION III. THEOREM. 370. Each angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon. B Let ABC, etc., be a regular polygon of n sides. § 180 AOB, BO C, etc., are equal, 4 rt. But the number of = n, the number of sides 371. COROLLARY. The radius drawn to any vertex of a regular polygon bisects the angle at that vertex. |