II O Geometry Pythagorean Triples (Sierpinski); Area = xy; more Pythagorean Triples per Sierpinski page and Proof of Right Triangle Area = xy “Every primitive Pythagorean triple (a, b, c) where b is an even integer is obtained only once per the following, where u and v are odd and u v: u v2 u v2 a uv b c 2 Here, u and v represent all pairs of odd, relatively prime natural numbers.” Waclaw Sierpinski, Pythagorean Triangles, The Scripta Mathematics Studies Number Nine New York City: Graduate School of Science, Yeshiva University, 1962 **************************************************************** Proof of Right Triangle Area = xy Dr Stan Hartzler Archer City High School x x y r r r Area = y 1 r perimeter = r (2r x y) r ( r x y ) 2 ( x r )2 ( y r )2 ( x y)2 x xr r y yr r x xy y 2 xr r yr r xy xr 2r yr xy xr r yr xy r ( x r y ) xy AREA II O Geometry Pythagorean Triples (Sierpinski); Area = xy; more Fun Relationships r a b c v (u v ) 2 x xy y ( x y ) v uv x ar 2 u uv y br a xr x xy y x y b yr x xy y x y c x y Perimeter = a b c 2(r x y ) x y x xy y uv u Area = ab u v uv xy Developing u, v for a given r Example: r = × × = 105 r uv v v 2r v 210 = u v v v 15 21 35 105 u 211 73 47 37 29 31 41 107 a 211 219 235 259 435 651 1435 11235 b 22260 2660 1092 660 308 260 228 212 c 22261 2669 1117 709 533 701 1453 11237 The number of distinct values of (u, v) for a given r of n odd prime factors is 2n And note that uk vk u2n 1k v2n 1k II O Geometry Pythagorean Triples (Sierpinski); Area = xy; more ... values of (u, v) for a given r of n odd prime factors is 2n And note that uk vk u2n 1k v2n 1k II O Geometry Pythagorean Triples (Sierpinski) ; Area = xy; more ... 35 105 u 211 73 47 37 29 31 41 107 a 211 219 235 259 435 651 1435 11235 b 22260 2660 109 2 660 3 08 260 2 28 212 c 22261 2669 1117 709 533 701 1453 11237 The number of distinct values of (u, v) for...II O Geometry Pythagorean Triples (Sierpinski) ; Area = xy; more Fun Relationships r a b c v (u v ) 2 x xy y ( x y )