A simple example is a triangle with side lengths:

3, 4 and 5

since

3

^{2}+ 4

^{2}= 9 + 16 = 25 = 5

^{2}

Some other examples are:

5, 12, 13 | (25 + 144 = 169) |

6, 8, 10 | (36 + 64 = 100) |

7, 24, 25 | (49 + 576 = 625) |

20, 21, 29 | (400 + 441 = 841) |

3, 4, 5

5, 12, 13

7, 24, 25

The first numbers in each row are the odd numbers 3, 5, 7 ...

To find the second number add the first number to the first two numbers in the previous row. So for the row starting with 5, add this 5 to the 3 and 4 from the previous row: 5 + 3 + 4 = 12. To get the final number just add one to the second number: 12 + 1 = 13.

Let's work out the next row. The odd number after 7 is 9:

9

Add 9 to the first two numbers in the previous row to get the second number: 9 + 7 + 24 = 40

9, 40

And add one more to get the final number is 40 + 1 = 41

9, 40, 41

(9

^{2}+ 40

^{2}= 81 + 1600 = 1681 = 41

^{2})

Another way to get the second and third numbers is to square the first number and halve the result. The second and third numbers are the whole numbers on either side. So for this last row: 9

^{2}= 81. Half of this is 40.5. So the second and third numbers are 40 and 41.

This method does not generate every possible Pythagorean Triangle. I'll leave that for another time.

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