spheres tangent to the planes of the faces of a tetrahedron

Notations: in the tetrahedron ABCD we denote a, b, c and d the areas of the faces opposite to A, B, C and D, and V its volume.

Besides the insphere (in green), tangent to the faces, there are also four exspheres (in orange), tangent to one face and to the planes of the three others.
More curious, there are also three other spheres called "attic spheres" (in red); they are tangent to the four planes of the faces and located in roof shaped alveolus (thus their name), the top of the attic being an edge of the tetrahedron.


regular tetrahedron
insphere and four exspheres (no attic sphere)


a tetrahedron with the five in/exsphères
and the three attic spheres


Use the "f" key (display switch for the blue polygons) to well see the spheres.

The radius of the insphere is  3V/(a+b+c+d)  and the one of the exsphere opposite to D is  3V/(a+b+c-d).
The second case above is the most frequent, nevertheless there are tetrahedra with only one or two attic spheres. According to David Pigeon (see reference) the existence of these spheres is linked to the areas of the tetrahedron's faces and their radii to the signs of the barycentric coordinates of their centers.
There cannot be spheres inscribed in two opposite attics (their radii would be opposite), thus there are at most three.

key property : If  a+b > c+d  then there is a tangent sphere with radius  rab=3V/(a+b-c-d)  in the attic with top AB.
Every inequality of this type leads to an attic sphere.

With for example  a ≥ b ≥ c ≥ d all the possible cases lead two five different configurations:
 •  a = b = c = d : NO attic sphere (this is the case of the equifacial tetrahedra)
 •  a = b > c = d : a+b > c+d  thus ONE sphere in the attic with top AB
 •  a ≥ b ≥ c > d  and  a+d = b+c :  a+b > c+d  and  a+c > b+d  thus TWO spheres in the attics with tops AB and AC
 •  a ≥ b ≥ c > d  and  a+d ≠ b+c:three inequalities  a+b > c+d ,  a+c > b+d  and ( a+d > b+c  ou  b+c > a+d ) thus THREE spheres in the attics with tops AB, AC and (AD or BC), and two different configurations
 •  a > b ≥ c = d : a+b > c+d ,  a+c > b+d  and  a+d > b+c  thus THREE spheres in the attics with tops AB, AC and AD, that is the first configuration above

Here are five other examples to illustrate the different configurations:




Thanks to my friend Nicolas Hannachi for having incited me to visit the attics of the tetrahedra, with images to support his argument.

equifacial tetrahedron (NO attic sphere)

tetrahedron with ONE attic sphere

tetrahedron with TWO attic spheres

regular tri-rectangular pyramid (THREE attic spheres)
the tops of the three attics have a common vertex

regular pointed pyramid (THREE attic spheres)
the tops of the three attics are the sides of a face


reference: Sphères des combles by David Pigeon - 2010 (in French)


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