PROPOSITION IV. THEOREM. 372. Two regular polygons of the same number of sides Let Q and Q' be two regular polygons, each having 2 rt. n sides. We are to prove Q and Q' similar polygons. The sum of the interior of each polygon is equal to (n - 2), § 157 (the sum of the interior of a polygon is equal to 2 rt. 4 taken as many times less 2 as the polygon has sides). 2 rt. 4 (n-2), Each of the polygon Q = n $158 (for the of a regular polygon are all equal, and hence each to the sum of the divided by their number). is equal .. the two polygons Q and Q' are mutually equiangular. .. the two polygons have their homologous sides proportional; .. the two polygons are similar. § 278 Q. E. D. PROPOSITION V. THEOREM. 373. The homologous sides of similar regular polygons have the same ratio as the radii of their circumscribed circles, and, also as the radii of their inscribed circles. etc. Let O and O' be the centres of the two similar regular polygons ABC, etc., and A'B'C', From O and O' draw O E, OD, O' E', O'D', also the Is Om and O'm'. O E and O' E' are radii of the circumscribed, and Om and O'm' are radii of the inscribed . $367 $368 the OED, ODE, O' E' D' and O' D' E' are equal, § 371 (being halves of the equal & FED, EDC, F' E' D' and E' D' C'); .. the AOED and O' E' D' are similar, § 280 (if two have two of the one equal respectively to two of the other, they are similar). Also, (the homologous sides of similar ▲ are proportional). $278 § 297 (the homologous altitudes of similar & have the same ratio as their homolo gous bases). Q. E. D. PROPOSITION VI. THEOREM. 374. The perimeters of similar regular polygons have the same ratio as the radii of their circumscribed circles, and, also as the radii of their inscribed circles. Let P and P' represent the perimeters of the two similar regular polygons ABC, etc., and A'B'C', etc. From centres O, O' draw O E, O' E', and is Om and O' m'. (the perimeters of similar polygons have the same ratio as any two homolo (the homologous sides of similar regular polygons have the same ratio as the (the homologous sides of similar regular polygons have the same ratio as PROPOSITION VII. THEOREM. 375. The circumferences of circles have the same ratio as their radii. Let C and C' be the circumferences, R and R' the radii of the two circles Q and Q'. Inscribe in the two regular polygons of the same number of sides. Conceive the number of the sides of these similar regular polygons to be indefinitely increased, the polygons continuing to be inscribed, and to have the same number of sides. Then the perimeters will continue to have the same ratio as the radii of their circumscribed circles, § 374 (the perimeters of similar regular polygons have the same ratio as the radii of their circumscribed ©), and will approach indefinitely to the circumferences as their limits. ... the circumferences will have the same ratio as the radii of their circles, $199 .. C: C' :: R : R'. Q. E. D. 376. COROLLARY. By multiplying by 2, both terms of the ratio R: R', we have CC: 2R: 2 R'; that is, the circumferences of circles are to each other as their diameters. That is, the ratio of the circumference of a circle to its diameter is a constant quantity. This constant quantity is denoted by the Greek letter π. 377. SCHOLIUM. The ratio is incommensurable, and therefore can be expressed only approximately in figures. The letter, however, is used to represent its exact value. Ex. 1. Show that two triangles which have an angle of the one equal to the supplement of the angle of the other are to each other as the products of the sides including the supplementary angles. 2. Show, geometrically, that the square described upon the sum of two straight lines is equivalent to the sum of the squares described upon the two lines plus twice their rectangle. 3. Show, geometrically, that the square described upon the difference of two straight lines is equivalent to the sum of the squares described upon the two lines minus twice their rectangle. 4. Show, geometrically, that the rectangle of the sum and difference of two straight lines is equivalent to the difference of the squares on those lines. |