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

= 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,
 * $$3\sqrt{3}R \geq a + b + c ,$$
 * $$ 9R^2 \geq a^2 + b^2 + c^2 ,$$
 * 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)