Answer to Question #21783 in Geometry for key amor t olivar

Question

Suppose that {a, b, c}is a pythagorean triple. Give some examples that will show that {3a+4b, 4a-3b, 5c} is pythagorean triple.

Answer 1

If {a, b, c}is a pythagoreantriple then:
a^2+b^2=c^2
And for {3a+4b, 4a-3b, 5c}:
(3a+4b)^2 + (4a-3b)^2 = (5c)^2
9a^2 + 12ab + 16b^2 + 16a^2 - 12ab + 9b^2 =25c^2
25a^2 + 25b^2 = 25c^2
a^2+b^2=c^2
It means that if {a, b, c}is a pythagorean triplethen {3a+4b, 4a-3b, 5c} is pythagorean triple too.