L'Huilier's theorem

L'Huilier's theorem is a theorem on a triangle in Euclidean geometry proved by the Swiss mathematician Simon Antoine Jean L'Huilier in 1809.

Theorem

L'Huilier's theorem—Let ${\displaystyle r}$ be the radius of the incircle of a triangle and ${\displaystyle r_{\text{A}},r_{\text{B}},r_{\text{C}}}$ be the radii of the three excircles. Then

${\displaystyle {\frac {1}{r}}={\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}}$

holds.^[1][2]

Proof

Let ${\displaystyle S}$ be the area of a triangle and ${\displaystyle a,b,c}$ be the lengths of the three sides. The reciprocal of the radius of the incircle is

${\displaystyle {\frac {1}{r}}={\frac {a+b+c}{2S}}}$

and the reciprocal of the radii of the excircles are

${\displaystyle {\begin{aligned}{\frac {1}{r_{\text{A}}}}&={\frac {-a+b+c}{2S}},\\{\frac {1}{r_{\text{B}}}}&={\frac {a-b+c}{2S}},\\{\frac {1}{r_{\text{C}}}}&={\frac {a+b-c}{2S}}.\end{aligned}}}$

Therefore, the sum of the reciprocals are

${\displaystyle {\begin{aligned}{\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}&={\frac {-a+b+c}{2S}}+{\frac {a-b+c}{2S}}+{\frac {a+b-c}{2S}}\\&={\frac {a+b+c}{2S}}={\frac {1}{r}}.\end{aligned}}}$

Extension

Although L'Huilier's theorem is a result on the Euclidean plane (two dimension), it can be extended to ${\displaystyle n}$-dimensional Euclidean space.

Let ${\displaystyle K}$ be an ${\displaystyle n}$-simplex (triangle in two-dimension and tetrahedron in three-dimension). The inscribed sphere can be defined as the sphere whose center is the point in the interior of ${\displaystyle K}$ that has equal distance to each face of ${\displaystyle K}$; let ${\displaystyle r_{0}}$ be its radius. Similarly, an escribed sphere can be defined as the sphere whose center is the point in the region to the opposite side of only one of the faces and has equal distance to each face. Because ${\displaystyle K}$ has ${\displaystyle n+1}$ faces, let these radii be ${\displaystyle r_{1},\ldots ,r_{n+1}}$. Then

${\displaystyle {\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}+\cdots +{\frac {1}{r_{n+1}}}={\frac {n-1}{r_{0}}}}$

holds.^[3] The proof uses linear algebra.

Related items

In his book, L'Huilier (1809) also suggested

${\displaystyle S={\sqrt {r\cdot r_{\text{A}}\cdot r_{\text{B}}\cdot r_{\text{C}}{\vphantom {A}}}}.}$

Since

${\displaystyle r_{\text{A}}\,r_{\text{B}}\,r_{\text{C}}={\frac {S^{2}}{r}}}$

holds,^[4] by multiplying to L'Huilier's theorem

${\displaystyle {\frac {1}{r_{\text{A}}}}+{\frac {1}{r_{\text{B}}}}+{\frac {1}{r_{\text{C}}}}={\frac {1}{r}},}$

we obtain

${\displaystyle r_{\text{B}}\,r_{\text{C}}+r_{\text{C}}\,r_{\text{A}}+r_{\text{A}}\,r_{\text{B}}=s^{2},}$

where ${\displaystyle s=(a+b+c)/2}$ is half of the circumference of the triangle.^[5][6]