The Incentre of a Tetrahedron

 Finding the Incentre (and Excentres) of a

 Tetrahedron using Geometric arguments

This is something I thought of mid-2020 during my JEE prep. Did not publish it then due to lack of time/ideas where to share it. I am going to assume things like the existence (and uniqueness) of incentres and excentres, which I think should be true for all tetrahedrons.

Notation : 

Consider a tetrahedron (the bounded region enclosed by 4 non-parallel planes in ℝ³, such that any 3 of these planes intersect at a distinct point, basically an arbitrary triangular pyramid). Label each of the vertices as 1,2,3,4, and the opposite plane face to each vertex with the same number as the vertex. I.e, plane "i" is opposite to vertex "i".

The Conventional Approach

If you are familiar with Analytical (or "Coordinate") Geometry, this problem should look fairly straightforward - solve the following system in (x,y,z,r) :

abs(aᵢx + bᵢy + cᵢz +dᵢ)/{(aᵢ² + bᵢ² + cᵢ²)^½} = r                                     for i = 1,2,3,4

Where aᵢx + bᵢy + cᵢz +dᵢ = 0 is the equation of Plane "i" forming the face of a Tetrahedron.

Intuitively, there must be 5 solutions to this system, as this simply gives the coordinates of a point equidistant from 4 planes (and the distance from each plane), a condition which is satisfied by the incentre as well as each excentre.

This method :

Let the position vector of the incentre be I, and that of the vertices be Vᵢ (i = 1,2,3,4). Let n̂_j be the outward unit normal vector to the plane face "j".

I state the following : 

∃ λ_i ∈ ℝ such that 

I - V_i  = λ_i{∑_cyclic (n̂_j × n̂_k)}

That is,  I - V₁ = λ₁ ( n̂₂ × n̂₃ + n̂₃ × n̂₄ + n̂₄ × n̂₂ ) , etc.

Let  I  =  F(i) :=  V_i  +  λ_i{∑_cyclic (n̂_j × n̂_k)}

Write this expression for 2 vertices, solve for one of the parameters (by saying F(a) = F(b), you will be able to determine the values of the scalars λ_a , λ_b) then plug the value of that parameter into the corresponding expression to find I.             

(given the vertices/plane faces, the n̂ vectors can be determined easily)

Proof : 

Kindly go through these, and do comment if something is unclear.

Thanks for reading, and do let me know if this/something similar is already known to you!

- Chinmay Giridhar