Inradius in Terms of Circumradius
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Theorem
Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $r$ denote the inradius of $\triangle ABC$.
Let $R$ denote the circumradius of $\triangle ABC$.
Then:
- $r = 4 R \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2$
Proof
Let $D$, $E$ and $F$ be the points where the incircle is tangent to the sides $AC$, $AB$ and $CB$ respectively.
Let $s$ denote the semiperimeter of $\triangle ABC$.
From Tangent Points of Incircle in Terms of Semiperimeter:
\(\ds AD\) | \(=\) | \(\ds s - a\) | ||||||||||||
\(\ds BE\) | \(=\) | \(\ds s - b\) | ||||||||||||
\(\ds CF\) | \(=\) | \(\ds s - c\) |
Then:
\(\ds r\) | \(=\) | \(\ds AI \sin \dfrac A 2\) | Definition of Sine of Angle | |||||||||||
\(\ds \dfrac {AI} {\sin \frac B 2}\) | \(=\) | \(\ds \dfrac c {\sin \angle AIB}\) | Law of Sines applied to $\triangle AIB$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac c {\map \sin {180 \degrees - \dfrac {A + B} 2} }\) | Sum of Angles of Triangle equals Two Right Angles | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac c {\sin \dfrac {A + B} 2}\) | Sine of Supplementary Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac c {\map \sin {\dfrac {180 \degrees - C} 2} }\) | Sum of Angles of Triangle equals Two Right Angles | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac c {\cos \frac C 2}\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 R \sin C} {\cos \frac C 2}\) | Law of Sines applied to $\triangle ABC$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 R \cdot 2 \sin \frac C 2 \cos \frac C 2} {\cos \frac C 2}\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 R \sin \frac C 2\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds AI\) | \(=\) | \(\ds 4 R \sin \dfrac B 2 \sin \frac C 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds r\) | \(=\) | \(\ds 4 R \sin \dfrac A 2 \sin \dfrac B 2 \sin \frac C 2\) |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(54)$