Wikipedia:Reference desk/Archives/Mathematics/2018 May 10

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May 10

Radius extrema for the circumcircle of a scalene triangle

What are the maxima and/or minima of the radius of the circumcircle of a scalene triangle with a, b, c as lengths of the sides? (Thanks)--82.137.9.60 (talk) 13:31, 10 May 2018 (UTC)

If side lengths a, b, c are given then the triangle and its circumcircle are uniquely defined (up to congruence), so the radius of the circumcircle has s single value. I don't see how "maxima and/or minima" applies here. Gandalf61 (talk) 15:04, 10 May 2018 (UTC)

I think the OP’s question may mean to choose a, b, and c so as to minimize the circumradius R. According to Circumscribed circle#Other properties,

$3{\sqrt {3}}R\geq a+b+c,$

$9R^{2}\geq a^{2}+b^{2}+c^{2},$

so in each of these the case of equality gives a lower bound on R. Loraof (talk) 17:13, 10 May 2018 (UTC)

And from that article, the exact expression is

circumradius = ${\frac {abc}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}}$

Loraof (talk) 17:27, 10 May 2018 (UTC)