More Pythagorean Triangles

Here's another rule for generating Pythagorean trianges, Last time each triangle began with an odd number. This time we will use even numbers:

4, 3, 5	(16 + 9 = 25)
6, 8, 10	(36 + 64 = 100)
8, 15, 17	(64 + 225 = 289)

You generate first number of the next line by writing down the next even number. To find the second and third numbers, add the first number to the second number in the previous row. The second and third numbers are one before and one after the result. For example, in the second row above, the first number is 6 as it is the even number that comes after the first number in the row above it, which is 4. We then add this 6 to the second number in the row above, which is 3. This gives us 9. The second and third numbers are one either side of this: 8 and 10.

Let's find the next row. The last row begins with 8. The even number after 8 is 10. So this is the first number:

10

Add this 10 to the second number of the previous row, which is 15. 10 + 15 = 25. The second and third numbers are one either side of this: 24 and 26

10, 24, 26

10² + 24² = 100 + 576 = 676 = 26²

Generate Pythagoras Triangles

Triangles containing a right angle are called Pythagorean triangles. Their side lengths have a simple relationship. The sum of the squares of the shorter side lengths equals the square of the length of the longer side.

A simple example is a triangle with side lengths:

3, 4 and 5

since

3² + 4² = 9 + 16 = 25 = 5²

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)

You can find some of these yourself as they fall into patterns. For example:

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

(9² + 40² = 81 + 1600 = 1681 = 41²)

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: 9² = 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.