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Given an arbitrary triangle where one of its sides is $a$ and the angle opposite to it is $A$, is there a circle with a unique radius $r$ such that this triangle is inscribed within it?

Omar Nagib
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    Yes, this is a consequence of a standard result in plane geometry. If nobody else explains in an answer, I’ll do that. – Lubin Oct 30 '15 at 22:34
  • The nearly universal convention for triangles is to use $r$ for the radius of the inscribed circle and $R$ for the radius of the circumcircle. – Mark Bennet Oct 30 '15 at 22:41

3 Answers3

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Yes: this is the inscribed angle theorem. See Wikipedia.

Bernard
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You can use the extended version of the sine rule to compute the radius of the circle $$\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}=2R$$ where $R$ is the circumradius of the triangle - this gives you $R$ in terms of the length of a side and the sine of the opposite angle, and that is the information you have to hand.

On the other hand, unless the angle given is a right-angle, there are two circles which conform to the criteria - if the given side is horizontal, one centre lies above the line and the other lies symmetrically below it.

Mark Bennet
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Consider the triangle formed by connecting the centre of the circumcircle to the endpoints of the line segment of length $a$

The inscribed angle theorem tells you that the angle between the radii is $2A$

WW1
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