# Radii of the Inscribed and Escribed Spheres of a Simplex

Published in International Journal of Geometry, 2014

In the picture below, $$A_1A_2A_3$$ is a triangle, $$I_0$$ is the center of the inscribed circle with radius $$r_0$$, and $$I_1,I_2,I_3$$ are the centers of the escribed circles with radii $$r_1,r_2,r_3$$. It is known that $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r_0}.$ This equation appears as (24) in Mackay (1893), who attributes to Steiner and Bobillier in 1828.

When I attended middle and high schools in the 1990s, I was fascinated by elementary geometry. I recall I discovered the above relation in 1997. In 1998, when I was a freshman at University of Tokyo, I found the three-dimensional generalization $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}+\frac{1}{r_4}=\frac{2}{r_0},$ where $$r_0$$ is the radius of the inscribed sphere of a tetrahedron and $$r_1,r_2,r_3,r_4$$ are the radii of the four escribed spheres. Based on some geometric intuition I conjectured that in the $$n$$-dimensional Euclidean space, the relation $\sum_{k=1}^{n+1}\frac{1}{r_k}=\frac{n-1}{r_0}$ would hold. After studying linear algebra in 1999, I proved this analytically.

In those days I didnâ€™t know how to publish papers, so my notes became dormant. After learning how to write papers and obtaining an academic job in 2013, I managed to publish it.

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