數學知識交流---勾股弦關係

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)因 a² + b² = c² , 故
(a + b + c)² = 2(c² + ab + bc + ca)
(a + b + c)² = 2(c(c + b) + a(c + b))
(a + b + c)² = 2 (c + a) (c + b)
a + b + c = ± √ ( 2 (c + a) (c + b) ) ........... ☆
由 ☆ :

a = ± √ ( 2 (c + a) (c + b) ) - (c + b) ...........A b = ± √ ( 2 (c + a) (c + b) ) - (c + a) ...........B c = ± √ ( 2 (c + a) (c + b) ) - a - b
- c = 干√ ( 2 (c + a) (c + b) ) + a + b

c = 干 √ ( 2 (c + a) (c + b) ) + (c + a) + (c + b) .......C
1) A 令 a , b 為負,
- a = ± √ ( 2 (c - a) (c - b) ) - (c - b)
a = 干√ ( 2 (c - a) (c - b) ) + (c - b)
a = 干√ ( 2 (27) (7233624) ) + (7233624) = 7213860 或 7253388同理,
b = 干√ ( 2 (c - a) (c - b) ) + (c - a)
b = 干√ ( 2 (27) (7233624) ) + (27) = - 19737(捨) 或 19791C 令 a , b 為負,
c = 干√ ( 2 (c - a) (c - b) ) + (c - a) + (c - b)
c = 干√ ( 2 (27) (7233624) ) + (27) + (7233624) = 7213887 或 7253415答 : a = 7 253 388 , b = 19791 , c = 7 253 415
2) A 令 b 為負,
a = ± √ ( 2 (c + a) (c - b) ) - (c - b)
a = ± √ ( 2 (c + a) ((c + a) - (a + b)) ) - ((c + a) - (a + b))
a = ± √ ( 2 (43520409) ((43520409) - (21779991)) ) - ((43520409) -(21779991))
a = 21760200 或 - 65241036 (捨)b = 21779991 - 21760200 = 19791c = 43520409 - 21760200 = 21760209 答 : a = 21 760 200 , b = 19791 , c = 21 760 209
3)C 令 a 為負 :
c = 干√ ( 2 (c - a) (c + b) ) + (c - a) + (c + b)
c = 干√ ( 2 (c - a) ((c - a) + (a + b)) ) + (c - a) + ((c - a) + (a + b))
c = 干√ ( 2 (3) ((3) + (65300403)) ) + (3) + ((3) + (65300403))
c = 65320203 或 65280615a = 65320200 或 65280612b = - 19797(捨) 或 19791答 : a = 65 280 612 , b = 19791 , c = 65 280 615
4)
由A :
a = ± √ ( 2 (c + a) (c + b) ) - (c + b)
a = ± √ ( 2 (391683681) (195861632) ) - (195861632)
a = 195841840 或 - 587565104 (捨) c = 391683681 - 195841840 = 195841841b = 195861632 - 195841841 = 19791答 : a = 195 841 840 , b = 19791 , c = 195 841 841

利用 a^2 + b^2 = c^2
若 c - a = x, c - b = y, 則
(c - x)^2 + (c - y)^2 = c^2
==> c^2 - 2(x + y)c + (x^2 + y^2) = 0
==> c = (x + y) + √(2xy) or c = (x + y) - √(2xy) [捨去, 此數小於 x 或 y]
==> c = (x + y) + √(2xy), a = y + √(2xy), b = x + √(2xy)

2012-02-27 21:18:17 補充：
1. c - a = 27, c - b = 7233624, 求 a, b, c。
設 x = 27, y = 7233624, 則 √(2xy) = 19764
所以 c = (27 + 7233624) + 19764, a = 7233624 + 19764, b = 27 + 19764
答案(1) : a = 7253388, b = 19791, c = 7253415