求 a : b : c

a/(c - b) + b/(c - a) = 6.5
a(c - a) + b(c - b)
─────────── = 13 / 2
　(c - b) (c - a)
c(a + b) - (a² + b²)
─────────── = 13 / 2
c² - c(a + b) + ab
　c(a + b) - c²
────────── = 13 / 2
c² - c(a + b) + ab
2 (c(a + b) - c²) = 13(c² - c(a + b)) + 13ab15c² - 15c(a + b) + 13ab = 0令 c = 1 , 則 a² + b² = c² = 1 , 且 15 - 15(a + b) + 13ab = 0.
(15 + 13ab)² = 225(a + b)²
169(ab)² + 390ab + 225 = 225(a² + b² + 2ab)
169(ab)² + 390ab + 225 = 225(1 + 2ab)
169(ab)² - 60ab = 0
ab = 60/169 = (5/13) (12/13)
而 (5/13)² + (12/13)² = 1 滿足 a² + b² = 1 ,
不妨設 a < b 則 a = 5/13 , b = 12/13 , c = 1.
∴ a : b : c = 5 : 12 : 13.


2015-03-05 20:13:00 補充：
或 a : b : c = 12 : 5 : 13.
　 ─────────────