Excircles and Exradii

10.8 Excircles and Exradii

Every triangle has, besides its incircle, three excircles — each tangent to one side and to the extensions of the other two. The excircle opposite $A$ has radius $r_1$, and similarly $r_2,r_3$ opposite $B,C$.

Theorem 1. $r_1=\dfrac{\Delta}{s-a}$.

Proof. The excentre $I_1$ opposite $A$ lies on the internal bisector of $A$ and the external bisectors of $B,C$, at distance $r_1$ from line $BC$ and from the extensions of $AB,AC$. Now $\Delta_{ABC}=\Delta_{ABI_1}+\Delta_{ACI_1}-\Delta_{BCI_1}$ (the triangle $BCI_1$ is subtracted since $I_1$ lies on the far side of $BC$ from $A$). Each of these three triangles has height $r_1$ on base $c,b,a$ respectively, so $\Delta=\frac12cr_1+\frac12br_1-\frac12ar_1=\frac{r_1}2(b+c-a)=r_1(s-a)$, giving $r_1=\Delta/(s-a)$.

Theorem 2. $r_1=s\tan\dfrac A2$.

Proof. The tangent length from $A$ to the excircle opposite $A$ equals $s$ (a standard tangent-length result). In the right triangle formed by $A$, the touch point on line $AB$ extended, and $I_1$, the angle at $A$ is again $\frac A2$, the adjacent side is $s$, and the opposite side is $r_1$. So $\tan\frac A2=r_1/s$, i.e. $r_1=s\tan\frac A2$.

Theorem 3. $r_1=4R\sin\dfrac A2\cos\dfrac B2\cos\dfrac C2$ (and cyclically for $r_2,r_3$), obtained the same way as §10.7 Theorem 3, by combining $r_1=s\tan\frac A2$ with the half-angle formulas and $s=4R\cos\frac A2\cos\frac B2\cos\frac C2$.

A key composite identity. Since $r=4R\sin\frac A2\sin\frac B2\sin\frac C2$ and $r_1=4R\sin\frac A2\cos\frac B2\cos\frac C2$ (and cyclically), adding all three exradii and subtracting $r$ collapses — via $\cos\frac{A}2\cos\frac B2-\sin\frac A2\sin\frac B2=\sin\frac C2$ and $\sin\frac A2\cos\frac B2+\cos\frac A2\sin\frac B2=\cos\frac C2$ (since $\frac A2+\frac B2=90^\circ-\frac C2$) — to the clean result $r_1+r_2+r_3-r=4R$ (proved in full in Exercise 10(b), Q6).

Worked example. For $a=13,b=14,c=15$: $s=21,\Delta=84,s-a=8$, so $r_1=\Delta/(s-a)=84/8=10.5$. This is the section Exercise 10(b) directly follows, since every inradius/exradius computation and identity in the exercise draws on §§10.7–10.8. …