English
The same as 117332, restated for a particular case: the preimage of the sphere under inversion equals the union of the center and the perpBisector of the inverted center.
Русский
То же, что и 117332, конкретно: предобраз сферы под инверсией равен объединению центра и перп биссектора инвертированного центра.
LaTeX
$$$ \operatorname{Set.preimage}(\mathrm{inversion}(c,R), \mathrm{sphere}(y, \operatorname{dist}(y,c))) = \{c\} \cup (\perpBisector(c, \mathrm{inversion}(c,R)\,y))$$$
Lean4
/-- The Monge point of a simplex (in 2 or more dimensions) is a
generalization of the orthocenter of a triangle. It is defined to be
the intersection of the Monge planes, where a Monge plane is the
(n-1)-dimensional affine subspace of the subspace spanned by the
simplex that passes through the centroid of an (n-2)-dimensional face
and is orthogonal to the opposite edge (in 2 dimensions, this is the
same as an altitude). The circumcenter O, centroid G and Monge point
M are collinear in that order on the Euler line, with OG : GM = (n-1): 2.
Here, we use that ratio to define the Monge point (so resulting
in a point that equals the centroid in 0 or 1 dimensions), and then
show in subsequent lemmas that the point so defined lies in the Monge
planes and is their unique point of intersection. -/
def mongePoint {n : ℕ} (s : Simplex ℝ P n) : P :=
(((n + 1 : ℕ) : ℝ) / ((n - 1 : ℕ) : ℝ)) • ((univ : Finset (Fin (n + 1))).centroid ℝ s.points -ᵥ s.circumcenter) +ᵥ
s.circumcenter
