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.
Proof
Let be the area of a triangle and be the lengths of the three sides. The reciprocal of the radius of the incircle isand the reciprocal of the radii of the excircles are
Therefore, the sum of the reciprocals are
Extension
Although L'Huilier's theorem is a result on the Euclidean plane, it can be extended to -dimensional Euclidean space.Let be an -simplex. The inscribed sphere can be defined as the sphere whose center is the point in the interior of that has equal distance to each face of ; let 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 has faces, let these radii be. Then
holds. The proof uses linear algebra.
Related items
In his book, L'Huilier also suggestedSince
holds, by multiplying to L'Huilier's theorem
we obtain
where is half of the circumference of the triangle.