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CTK Exchange
Vadim Tropashko
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Feb-02-04, 06:08 PM (EST) |
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"Pythagorean triple"
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Right triangle on https://www.cut-the-knot.org/proofs/fords.shtml is Pythagorean triple. The proof is obvious: the hypothenus is
a sum of radiuses, and therefore is rational number. One side is the differences of radiuses, and the other side is the difference between
neigbour farey fractions. Multiply all the 3 rationals to get inteder numbers, then. I wonder if we generate all the Pythagorean triples this way. |
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alexb
Charter Member
1194 posts |
Feb-02-04, 07:41 PM (EST) |
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1. "RE: Pythagorean triple"
In response to message #0
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>Right triangle on
>
>https://www.cut-the-knot.org/proofs/fords.shtml
>
>is Pythagorean triple. The proof is obvious: the hypothenus
>is a sum of radiuses, and therefore is rational number. One
>side is the differences of radiuses, and the other side is
>the difference between neigbour farey fractions. Multiply
>all the 3 rationals to get inteder numbers, then. I wonder
>if we generate all the Pythagorean triples this way. Yes, of course. After multiplication, the sides depend only on b and d in the right kind of a way, viz., d2 - b2, 2db, d2 + b2,. For the Ford circles to touch, we must have bc - ad = 1. But this means that b and d are coprime. If, on the other hand, b and d are coprime, then a and c exist such that bc - ad = 1, so that the circles corresponding to a/b and c/d touch.
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