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\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}

Source: Moldova TST 2004

March 8, 2023
geometry3D geometrytetrahedronsphereinequalities

Problem Statement

In the tetrahedron ABCDABCD the radius of its inscribed sphere is rr and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are rA,rB,rC,rD.r_A, r_B, r_C, r_D. Prove the inequality 1rA2−rArB+rB2+1rB2−rBrC+rC2+1rC2−rCrD+rD2+1rD2−rDrA+rA2≤2r.\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.
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