Abouabdillah's theorem
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Abouabdillah's theorem refers to two distinct theorems in mathematics: one in geometry and one in number theory.
[edit] Geometry
In geometry, similarities of an Euclidean space preserve circles and spheres. Conversely, the Abouabdillah's theorem states that every injective or surjective transformation of a Euclidean space that preserves circles or spheres is a similarity.
More precisely:
Theorem. Let E be a Euclidean affine space of dimension at least 3. Then:
1. Every surjective application 
that transforms any four concyclic points into four concyclic points is a similarity.
2. Every injective application that transforms any circle into a circle is a similarity.
[edit] Number Theory
In number theory the Abouabdillah's theorem is about antichains of N. ( An antichain of N, for divisibility, is a set of non null integers such that no one is divisible by another. It possible to prove using Dilworth's theorem that the maximal cardinality of an antichain of E2n = {1,2,...,2n} is n).
Abouabdillah's Theorem. Let 
, c = 2kc', c' odd. Then E2n contains an antichain of cardinality n containing c if and only if 2n < 3k+1c'.
