Problem A sept 2011

Fix a point P in the interior of a face of a regular tetrahedron Δ. Show that Δ can be partitioned into four congruent convex polyhedra such that P is a vertex of one of them.

Solution:

Let A,B,C,D be the vertices of Δ and suppose P is an arbitrary point in the interior of face ABC.
P is determined by its distances λ, μ, ν to A,B,C resp.
Let us fix three more points so that on each face we have one of the four fixed points and each of A,B,C,D is at distance λ from one of them, at distance μ from another one, and at distance ν from a third one:

Let T_ABC be the point on face ABC at distances λ, μ, ν to A,B,C resp. (So T_ABC = P.)
Let T_BCD be the point on face BCD at distances λ, μ, ν to C,D,B resp.
Let T_CDA be the point on face CDA at distances λ, μ, ν to D,C,A resp.
Let T_DAB be the point on face DAB at distances λ, μ, ν to B,A,D resp.

So δ := T_ABCT_BCDT_CDAT_DAB is also a regular tetrahedron. Let O be its center.

Now we will allot points of Δ to each X ∈ {A,B,C,D} in such a way that the points allotted to X form a convex polyhedron, and such that the four polyhedra are congruent and form a partition of Δ.
The allotment will first be done in two steps:

Step 1)
The points in the hexahedron AT_ABCT_CDAT_DABO belong to A.
The points in the hexahedron BT_ABCT_BCDT_DABO belong to B.
The points in the hexahedron CT_ABCT_BCDT_CDAO belong to C.
The points in the hexahedron DT_BCDT_CDAT_DABO belong to D.

Step 2)
To allot the remaining points in the interior of Δ while preserving convexity and congruence, we extend (to the exterior of δ) the six planes through O and a side of δ that bisect the angles between two faces of δ.
Through each vertex of δ there go three of these bisectrix planes.
Now for each X ∈ {A,B,C,D}, allot to X all remaining points y ∈ Δ such that, for each plane through O and a side of the nearest face of δ, X and y are at the same side of that plane.

Of course, we can sum it all up, including both steps 1 and 2, as follows:
For each X ∈ {A,B,C,D}, allot to X all points y ∈ Δ such that, for each plane through O and a side of the nearest face of δ, X and y are at the same side of that plane.

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