Generating Pythagorean Triples

There is a simple formula that gives all the Pythagorean triples.
Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple. It's easy to check algebraically that the sum of the squares of the first two is the same as the square of the last one. Why is it that every triple can be generated in this manner? Here are the first few triples for m and n between 1 and 10.

Notice any patterns? m= 1 2 3 4 5 6 7 8 9 ---+------------------------------------------------------------------------------------------------------------------ n= 2 [3,4,5] 3 [8,6,10] [5,12,13] 4 [15,8,17] [12,16,20] [7,24,25] 5 [24,10,26] [21,20,29] [16,30,34] [9,40,41] 6 [35,12,37] [32,24,40] [27,36,45] [20,48,52] [11,60,61] 7 [48,14,50] [45,28,53] [40,42,58] [33,56,65] [24,70,74] [13,84,85] 8 [63,16,65] [60,32,68] [55,48,73] [48,64,80] [39,80,89] [28,96,100] [15,112,113] 9 [80,18,82] [77,36,85] [72,54,90] [65,72,97] [56,90,106] [45,108,117] [32,126,130] [17,144,145] 10 [99,20,101] [96,40,104] [91,60,109] [84,80,116] [75,100,125] [64,120,136] [51,140,149] [36,160,164] [19,180,181]