Here are some more Pythagorean triples: [6, 8, 10] [9, 12, 15] [12, 16, 20] [15, 20, 25] [18, 24, 30] [21, 28, 35]

Can you see how we created these? Can you create the next triple in the sequence?

Here is another sequence of Pythagorean triples:

[5, 12, 13] , [7, 24, 25] , [9, 40, 41] , [11, 60, 61] , [12, 35, 37] , [20, 21, 29], [20, 99, 101], [60, 91, 109], [15, 112, 113],
[47, 117, 125], [88, 105, 137], [17, 144, 145], [24, 143, 145]

Can you see how we created these? Can you create the next triple in the sequence?

Can you make more Pythagorean triples?

In Circle

Notice that when the triangle is defined by any of the triples above, the radius of the incircle is an integer.

Here is the formula for the radius of the incircle of a triangle whose sides are length a,b,c:




This is closely related to Heron's formula for the area of a circle and can be rewritten like this:




Verify that this formula gives an integer for the Pythagorean triples shown above.

Can you find any values for a, b, c which give an integer value for the incircle radius, but which are NOT a Pythagorean triple?

Can you prove that the incircle radius is an integer for all Pythagorean triples generated in the way shown above?

Can you prove that the incircle radius is an integer for all Pythagorean triples?