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Prove this inequality on radius of spheres of a tetrahedron

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020
tetrahedroncircumsphereinsphere3D geometryinequalitiesexsphere

Problem Statement

In the [ABCD][ABCD] tetrahedron having all the faces acute angled triangles, is denoted by rXr_X, RXR_X the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the X∈{A,B,C,D}X \in \{A,B,C,D\} peak, and with RR the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs8R2≥(rA+RA)2+(rB+RB)2+(rC+RC)2+(rD+RD)28R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2
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