Can a circle be specified by an arbitrary triangle when one of its sides and the angle opposite to it is known?

Question

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?

Answer 1

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

Answer 2

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.

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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.

Answer 3

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 $2A$