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 ...
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