Incircle and excircles

Excircles and excenters

An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.

The center of an excircle is the intersection of the internal bisector of one angle (at vertex ${\displaystyle A}$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex ${\displaystyle A}$, or the excenter of ${\displaystyle A}$. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.

Trilinear coordinates of excenters

While the incenter of ${\displaystyle \mathrm{△}ABC}$ has trilinear coordinates ${\displaystyle 1:1:1}$, the excenters have trilinears $${\displaystyle \begin{array}{rrcrcr}{J}_A=& -1& :& 1& :& 1\\ J_B=& 1& :& -1& :& 1\\ J_C=& 1& :& 1& :& -1\end{array}}$$

Exradii

The radii of the excircles are called the exradii.

The exradius of the excircle opposite ${\displaystyle A}$ (so touching ${\displaystyle BC}$, centered at ${\displaystyle J_A}$) is $${\displaystyle r_a=\frac{rs}{s-a}=\sqrt{\frac{s(s-b)(s-c)}{s-a}},}$$ where ${\displaystyle s=\frac{1}{2}(a+b+c).}$

See Heron's formula.

Derivation of exradii formula

Source:

Let the excircle at side ${\displaystyle AB}$ touch at side ${\displaystyle AC}$ extended at ${\displaystyle G}$, and let this excircle's radius be ${\displaystyle r_c}$ and its center be ${\displaystyle J_c}$. Then ${\displaystyle J_{c}G}$ is an altitude of ${\displaystyle \mathrm{△}AC{J}_c}$, so ${\displaystyle \mathrm{△}AC{J}_c}$ has area ${\displaystyle \frac{1}{2}b{r}_c}$. By a similar argument, ${\displaystyle \mathrm{△}BC{J}_c}$ has area ${\displaystyle \frac{1}{2}a{r}_c}$ and ${\displaystyle \mathrm{△}AB{J}_c}$ has area ${\displaystyle \frac{1}{2}c{r}_c}$. Thus the area ${\displaystyle \mathrm{\Delta }}$ of triangle ${\displaystyle \mathrm{△}ABC}$ is $${\displaystyle \mathrm{\Delta }=\frac{1}{2}(a+b-c)r_c=(s-c)r_c.}$$

So, by symmetry, denoting ${\displaystyle r}$ as the radius of the incircle, $${\displaystyle \mathrm{\Delta }=sr=(s-a)r_a=(s-b)r_b=(s-c)r_c.}$$

By the Law of Cosines, we have $${\displaystyle \cos A=\frac{b^2+c^2-a^2}{2bc}}$$

Combining this with the identity ${\displaystyle {\sin }^{2}A+{\cos }^{2}A=1}$, we have $${\displaystyle \sin A=\frac{\sqrt{-a^4-b^4-c^4+2a^2{b}^2+2b^2{c}^2+2a^2{c}^2}}{2bc}}$$

But ${\displaystyle \mathrm{\Delta }=\frac{1}{2}bc\sin A}$, and so $${\displaystyle \begin{array}{cc}\mathrm{\Delta }& =\frac{1}{4}\sqrt{-a^4-b^4-c^4+2a^2{b}^2+2b^2{c}^2+2a^2{c}^2}\\ & =\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\\ & =\sqrt{s(s-a)(s-b)(s-c)},\end{array}}$$ which is Heron's formula.

Combining this with ${\displaystyle sr=\mathrm{\Delta }}$, we have $${\displaystyle r^2=\frac{{\mathrm{\Delta }}^2}{s^2}=\frac{(s-a)(s-b)(s-c)}{s}.}$$

Similarly, ${\displaystyle (s-a)r_a=\mathrm{\Delta }}$ gives $${\displaystyle \begin{array}{cc}& r_a^2=\frac{s(s-b)(s-c)}{s-a}\\ & ⟹r_a=\sqrt{\frac{s(s-b)(s-c)}{s-a}}.\end{array}}$$

Other properties

From the formulas above one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields: $${\displaystyle \mathrm{\Delta }=\sqrt{r{r}_a{r}_b{r}_c}.}$$

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. The radius of this Apollonius circle is ${\displaystyle \frac{r^2+s^2}{4r}}$ where ${\displaystyle r}$ is the incircle radius and ${\displaystyle s}$ is the semiperimeter of the triangle.

The following relations hold among the inradius ${\displaystyle r}$, the circumradius ${\displaystyle R}$, the semiperimeter ${\displaystyle s}$, and the excircle radii ${\displaystyle r_a}$, ${\displaystyle r_b}$, ${\displaystyle r_c}$: $${\displaystyle \begin{array}{cc}{r}_a+r_b+r_c& =4R+r,\\ r_a{r}_b+r_b{r}_c+r_c{r}_a& =s^2,\\ r_a^2+r_b^2+r_c^2& ={\left(4R+r\right)}^2-2s^2.\end{array}}$$

The circle through the centers of the three excircles has radius ${\displaystyle 2R}$.

If ${\displaystyle H}$ is the orthocenter of ${\displaystyle \mathrm{△}ABC}$, then $${\displaystyle \begin{array}{cc}{r}_a+r_b+r_c+r& =\overline{AH}+\overline{BH}+\overline{CH}+2R,\\ r_a^2+r_b^2+r_c^2+r^2& ={\overline{AH}}^2+{\overline{BH}}^2+{\overline{CH}}^2+(2R{)}^2.\end{array}}$$

Nagel triangle and Nagel point

The Nagel triangle or extouch triangle of ${\displaystyle \mathrm{△}ABC}$ is denoted by the vertices ${\displaystyle T_A}$, ${\displaystyle T_B}$, and ${\displaystyle T_C}$ that are the three points where the excircles touch the reference ${\displaystyle \mathrm{△}ABC}$ and where ${\displaystyle T_A}$ is opposite of ${\displaystyle A}$, etc. This ${\displaystyle \mathrm{△}{T}_A{T}_B{T}_C}$ is also known as the extouch triangle of ${\displaystyle \mathrm{△}ABC}$. The circumcircle of the extouch ${\displaystyle \mathrm{△}{T}_A{T}_B{T}_C}$ is called the Mandart circle (cf. Mandart inellipse).

The three line segments ${\displaystyle \overline{A{T}_A}}$, ${\displaystyle \overline{B{T}_B}}$ and ${\displaystyle \overline{C{T}_C}}$ are called the splitters of the triangle; they each bisect the perimeter of the triangle, $${\displaystyle \overline{AB}+\overline{B{T}_A}=\overline{AC}+\overline{C{T}_A}=\frac{1}{2}\left(\overline{AB}+\overline{BC}+\overline{AC}\right).}$$

The splitters intersect in a single point, the triangle's Nagel point ${\displaystyle N_a}$ (or triangle center X₈).

Trilinear coordinates for the vertices of the extouch triangle are given by $${\displaystyle \begin{array}{cccccc}{T}_A=& 0& :& {\csc }^2\frac{B}{2}& :& {\csc }^2\frac{C}{2}\\ T_B=& {\csc }^2\frac{A}{2}& :& 0& :& {\csc }^2\frac{C}{2}\\ T_C=& {\csc }^2\frac{A}{2}& :& {\csc }^2\frac{B}{2}& :& 0\end{array}}$$

Trilinear coordinates for the Nagel point are given by $${\displaystyle {\csc }^2\frac{A}{2}:{\csc }^2\frac{B}{2}:{\csc }^2\frac{C}{2},}$$

or, equivalently, by the Law of Sines, $${\displaystyle \frac{b+c-a}{a}:\frac{c+a-b}{b}:\frac{a+b-c}{c}.}$$

Barycentric coordinates for the Nagel point are therefore $${\displaystyle b+c-a:c+a-b:a+b-c,}$$ or equivalently $${\displaystyle s-a:s-b:s-c.}$$

The Nagel point is the isotomic conjugate of the Gergonne point.

Related constructions

Nine-point circle and Feuerbach point

In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:

• The midpoint of each side of the triangle
• The foot of each altitude
• The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).

In 1822, Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that:

... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... (Feuerbach 1822)

The triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.

The incircle may be described as the pedal circle of the incenter. The locus of points whose pedal circles are tangent to the nine-point circle is known as the McCay cubic.

Incentral and excentral triangles

The points of intersection of the interior angle bisectors of ${\displaystyle \mathrm{△}ABC}$ with the segments ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$ are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle ${\displaystyle \mathrm{△}{A}^{\prime }{B}^{\prime }{C}^{\prime }}$ are given by $${\displaystyle \begin{array}{cccccc}{A}^{\prime }=& 0& :& 1& :& 1\\ B^{\prime }=& 1& :& 0& :& 1\\ C^{\prime }=& 1& :& 1& :& 0\end{array}}$$

The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page). Trilinear coordinates for the vertices of the excentral triangle ${\displaystyle \mathrm{△}{J}_A{J}_B{J}_C}$ are given by $${\displaystyle \begin{array}{rrcrcr}{J}_A=& -1& :& 1& :& 1\\ J_B=& 1& :& -1& :& 1\\ J_C=& 1& :& 1& :& -1\end{array}}$$

Equations for four circles

Let ${\displaystyle x:y:z}$ be a variable point in trilinear coordinates, and let ${\displaystyle u={\cos }^2\left(A/2\right)}$, ${\displaystyle v={\cos }^2\left(B/2\right)}$, ${\displaystyle w={\cos }^2\left(C/2\right)}$. The four circles described above are given equivalently by either of the two given equations:

• Incircle:$${\displaystyle \begin{array}{cc}{u}^2{x}^2+v^2{y}^2+w^2{z}^2-2vwyz-2wuzx-2uvxy& =0\\ \pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{y\phantom{t}}\cos \frac{B}{2}\pm \sqrt{z}\cos \frac{C}{2}& =0\end{array}}$$
• ${\displaystyle A}$-excircle:$${\displaystyle \begin{array}{cc}{u}^2{x}^2+v^2{y}^2+w^2{z}^2-2vwyz+2wuzx+2uvxy& =0\\ \pm \sqrt{-x}\cos \frac{A}{2}\pm \sqrt{y\phantom{t}}\cos \frac{B}{2}\pm \sqrt{z}\cos \frac{C}{2}& =0\end{array}}$$
• ${\displaystyle B}$-excircle:$${\displaystyle \begin{array}{cc}{u}^2{x}^2+v^2{y}^2+w^2{z}^2+2vwyz-2wuzx+2uvxy& =0\\ \pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{-y\phantom{t}}\cos \frac{B}{2}\pm \sqrt{z}\cos \frac{C}{2}& =0\end{array}}$$
• ${\displaystyle C}$-excircle:$${\displaystyle \begin{array}{cc}{u}^2{x}^2+v^2{y}^2+w^2{z}^2+2vwyz+2wuzx-2uvxy& =0\\ \pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{y\phantom{t}}\cos \frac{B}{2}\pm \sqrt{-z}\cos \frac{C}{2}& =0\end{array}}$$

Euler's theorem

Euler's theorem states that in a triangle: $${\displaystyle (R-r{)}^2=d^2+r^2,}$$ where ${\displaystyle R}$ and ${\displaystyle r}$ are the circumradius and inradius respectively, and ${\displaystyle d}$ is the distance between the circumcenter and the incenter.

For excircles the equation is similar: $${\displaystyle {\left(R+r_{\text{ex}}\right)}^2=d_{\text{ex}}^2+r_{\text{ex}}^2,}$$ where ${\displaystyle r_{\text{ex}}}$ is the radius of one of the excircles, and ${\displaystyle d_{\text{ex}}}$ is the distance between the circumcenter and that excircle's center.

Generalization to other polygons

Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. Among their many properties, perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the Pitot theorem.

More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a tangential polygon.

Generalization to topological triangles

If topological triangles are considered, it is still possible to define an inscribed circle. It is no longer described as tangent to all sides, since the topological triangle might not be differentiable everywhere. Rather, it is defined as a circle whose center has the same minimal distance to each side. It has been proven that all topological triangles have an inscribed circle.