A Note on Circumradius

For a triangle, its circumcircle is the circle that passes through the three vertices; the radius of the circle is called circumradius.

In my last post, I wrote about finding radius of incircle of a triangle. In this post, I will touch on circumradius.

Question: For a triangle $ABC$ with lengths of the three sides being $a,\ b$ and $c$, what is its circumradius?

Answer: Denote the circumradius by $r_C$. Let

$$ s = \frac{a+b+c}{2}. $$

According to the law of sines,

$$ 2r_C = \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{abc}{2T}, $$

where $T$ is the area of the triangle, and

$$ T = \sqrt{s(s-a)(s-b)(s-c)}. $$

Thus,

$$ r_C = \frac{abc}{4T} = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}. $$

Remark:

Denote the incircle radius by $r_I$. As shown in my last post,

$$ r_I = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}. $$

Note that

$$ r_C r_I = \frac{abc}{4s} = \frac{abc}{2(a + b + c)}. $$

Typing “the product of incircle radius and circumradius” into Google, I was pointed to the Wikipedia article Circumcircle. In the article, it clearly states that

The product of the incircle radius and the circumcircle radius of a triangle with sides $a$, $b$, $c$ is

$$ \frac{abc}{2(a + b + c)} $$

Very interesting!