Number is the ruler of forms and ideas, and the cause of gods and daemons.

— Pythagoras, as attributed by Iamblichus

This diagram shows (but not to scale ;-)) a clever method of constructing two Pythagorean triples given any two other Pythagorean triples due to Francois Viete.

Given (a, b, c, d, e, f are integers):

a^2 + b^2 = c^2
d^2 + e^2 = f^2

two other triples are found by synaeresis:

A = ae + bd
B = be – ad

and diaeresis:

D = ae – bd
E = be + ad

(I guess a, b, d, and e can always be picked so that B > 0 and D > 0, that is, be > ad and ae > bd?)

so that:

A^2 + B^2 = (cf)^2
D^2 + E^2 = (cf)^2

Voila!

Further Reading:

https://en.wikipedia.org/wiki/Pythagorean_triple

Special thanks to Pat’s Blog for this little gem:

https://pballew.blogspot.com/2018/10/viete-on-pythagorean-triples.html

Additionally, these relationships are also used in Fermat’s Theorem on sums of 2 squares

https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_theorem_on_sums_of_two_squares

[*9.138, *10.186, *10.187]

<>