數學知識交流---勾股弦關係

因為 a^2 + b^2 = c^2, 所以
a^2 = c^2 - b^2 = (c - b)(c + b) . . . (i)
b^2 = c^2 - a^2 = (c - a)(c + a) . . . (ii)
(a + b)^2 - c^2 = 2ab . . . . . . . . .(iii)
c^2 - (a - b)^2 = 2ab . . . . . . . . .(iv)
1. a = 19791, b + c = 59373, 求 b, c.
從 (i) 得 c - b = 19791^2 / 59373 = 6597
所以 c = (59373 + 6597)/2 = 32985, b = 59373 - 32985 = 26388
答案(1) : b = 26388, c = 329852. b = 87960, a + c = 109950, 求 a, c.
從 (ii) 得 c - a = 87960^2 / 109950 = 70368
所以 c = (109950 + 70368)/2 = 90159, a = 109950 - 90159 = 19791
答案(2) : a = 19791, c = 901593. a = 266812, c - b = 247754, 求 b, c.
從 (i) 得 c + b = 266812^2 / 247754 = 287336
所以 c = (247754 + 287336)/2 = 267545, b = 267545 - 247754 = 19791
答案(3) : b = 19791, c = 2675454. b = 268280, c - a = 249218, 求 a, c.
從 (ii) 得 c + a = 268280^2 / 249218 = 288800
所以 c = (249218 + 288800)/2 = 269009, a = 269009 - 249218 = 19791
答案(4) : a = 19791, c = 2690095. c = 806055, a + b = 825603, 求 a, b.
從 (iii) 得 ab = (825603^2 - 806055^2)/2
= (825603 + 806055)(825603 - 806055)/2
= 815829 * 19548
= 19791 * 805812 . . . . (19791 + 805812 = 825603)
所以 a = 19791, b = 805812 或 a = 805812, b = 19791
答案(5) : a = 19791, b = 805812 或 a = 805812, b = 197916. c = 2417841, a - b = 2397969, 求 a, b.
從 (iii) 得 ab = (2417841^2 - 2397969^2)/2
= (2417841 + 2397969)(2417841 - 2397969)/2
= 2407905 * 19872
= 19791 * 2417760 . . . . (2417760 - 19791 = 2397969)
所以 a = 2417760, b = 19791
答案(6) : a = 2417760, b = 19791(ps. 有趣, 每個三角形都有同一個數字 19791.)