English
Within a cospherical subset of a subspace, two simplices have equal circumcenter.
Русский
В пределах кососферного подмножества подпространства два симплекса имеют одинаковый circumcenter.
LaTeX
$$$\exists c \in P, ∀ sx : \text{Simplex } \mathbb{R} P n, (\text{range } sx.points ⊆ ps) ⇒ sx.circumcenter = c$$$
Lean4
/-- **Side–Angle–Side (SAS) congruence**
If two triangles have two sides and the included angle equal, then the triangles are congruent.
This holds even if the triangles are degenerate. -/
theorem side_angle_side (h : ∠ a b c = ∠ a' b' c') (hd₁ : dist a b = dist a' b') (hd₂ : dist b c = dist b' c') :
![a, b, c] ≅ ![a', b', c'] := by
apply side_side_side hd₁ hd₂
rw [dist_comm, dist_comm c' a', ← sq_eq_sq₀ (by positivity) (by positivity), pow_two, pow_two,
EuclideanGeometry.law_cos a b c, EuclideanGeometry.law_cos a' b' c']
simp [h, hd₁, hd₂, dist_comm]
