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Three consecutive terms of a geometric series a, b, c have a common ratio `b/a = c/b = r` .
If the three sides of a right triangle a, b, c form a geometric series, the length of the sides can be expressed as a, ar and ar^2.
The sides of the triangle follow the rule `a^2 + (ar)^2 = (ar^2)^2`
=> `1 + r^2 = r^4`
=> `r^4 - r^2 - 1 = 0`
Let `r^2 = x`
=> `x^2 - x - 1 = 0`
=> `x = (1 +- sqrt(1 + 4))/2 = (1 +- sqrt 5)/2`
For `x = (1 + sqrt 5)/2` , r is real.
It is possible to have three sides of a right triangle form a geometric series.
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