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Subject: "Ford's touching circles"

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Akira Bergman

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Feb-24-06, 08:04 AM (EST)

"Ford's touching circles"

Is there a mistake at;
https://www.cut-the-knot.org/proofs/fords.shtml
I can not fill in the proof that bc-ad=1;

AC^2 = AB^2 + BC^2
<(1/b^2)+(1/d^2)>^2 = <(1/b^2)-(1/d^2)>^2 + <(c/d)-(a/b)>^2
<(b^2+d^2)/(bd)^2>^2 = <(b^2-d^2)/(bd)^2>^2 + <(bc-ad)/bd>^2
4(bd)^2/(bd)^4 = (bc-ad)^2/(bd)^2
(bc-ad)^2=4
?

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alexb
Charter Member
1783 posts

Feb-24-06, 09:31 AM (EST)

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1. "RE: Ford's touching circles"
In response to message #0

>Is there a mistake at;
>https://www.cut-the-knot.org/proofs/fords.shtml
>I can not fill in the proof that bc-ad=1;
>
>AC^2 = AB^2 + BC^2
><(1/b^2)+(1/d^2)>^2 = <(1/b^2)-(1/d^2)>^2 + <(c/d)-(a/b)>^2

You had to write

BC^2 = AB^2 + AC^2
<(1/b^2)+(1/d^2)>^2/4 = <(1/b^2)-(1/d^2)>^2/4 + <(c/d)-(a/b)>^2

For, as stated,

2·BC = 1/b² + 1/d² and
2·AB = 1/b² - 1/d².

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