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/-!
# Problem 2: Algebraic certificate for `OM = ON`
This file formalises the polynomial core of the coordinate proof in Lean 4.
The theorem `polynomial_certificate` is completely proved by `ring`.
The theorem `algebraic_finish` proves the distance equality from the cleared
angle equations and the two circumcentre equations.
A fully synthetic encoding of Euclidean angles and the interior-point
hypotheses is intentionally separated from this algebraic certificate.
-/
set_option autoImplicit false
namespace Problem2
/-- Polynomial arising from the two angle conditions. -/
def F (a c s t : β) : β :=
c * ((1 + 2 * a^2) * t^2 + a^2) +
s * (a * (t^2 + 1) - (a^2 + 1) * t)
def Hq (a q Ο c s : β) : β :=
Ο * (a + q)^2 - F a c s q
def Hp (a p Ο c s : β) : β :=
Ο * F a c s p - (a + p)^2
def H0 (c s : β) : β :=
c^2 + s^2 - 1
def Delta (p q c s : β) : β :=
s * (1 - p*q) - c * (p + q)
def Den (a p q c s : β) : β :=
(a + p) * (a + q) * Delta p q c s
def CenterRHS (a p q Ο c s : β) : β :=
2*a *
(Ο * (1 + p^2) * (a + q) * (Ο * (s - q*c) + q) -
(1 + q^2) * (a + p) * (s + p * (Ο - c)))
def T (a p q Ο c s : β) : β :=
CenterRHS a p q Ο c s -
(Ο^2 - 1) * Den a p q c s
def Up (a p Ο c s : β) : β :=
a + p + Ο * (-2*a*c*p^2 - a*c + a*p*s + c*p + p^2*s)
def Vq (a q Ο c s : β) : β :=
Ο * (a + q) - 2*a*c*q^2 - a*c + a*q*s + c*q + q^2*s
def W (a p q : β) : β :=
2*a^2*p*q - a^2 - a*p - a*q + p*q
/-- The exact polynomial certificate used in the written solution. -/
theorem polynomial_certificate
(a p q Ο c s : β) :
T a p q Ο c s =
Up a p Ο c s * Hq a q Ο c s +
Vq a q Ο c s * Hp a p Ο c s -
Ο * (p - q) * W a p q * H0 c s := by
unfold T CenterRHS Den Delta Up Vq W Hq Hp H0 F
ring
/--
Algebraic completion of the proof.
`hcenter` is the cleared-denominator form obtained by solving the two linear
circumcentre equations. `hHq` and `hHp` are the two cleared angle equations.
-/
theorem algebraic_finish
(a p q Ο c s u v : β)
(hHq : Hq a q Ο c s = 0)
(hHp : Hp a p Ο c s = 0)
(hunit : c^2 + s^2 = 1)
(hcenter :
4 * ((Ο - c) * u - s*v) * Den a p q c s =
CenterRHS a p q Ο c s)
(hden : Den a p q c s β 0) :
(u - Ο/2)^2 + v^2 =
(u - c/2)^2 + (v - s/2)^2 := by
have hH0 : H0 c s = 0 := by
unfold H0
linarith
have hT : T a p q Ο c s = 0 := by
calc
T a p q Ο c s =
Up a p Ο c s * Hq a q Ο c s +
Vq a q Ο c s * Hp a p Ο c s -
Ο * (p - q) * W a p q * H0 c s :=
polynomial_certificate a p q Ο c s
_ = 0 := by rw [hHq, hHp, hH0]; ring
have hmul :
(4 * ((Ο - c) * u - s*v) - (Ο^2 - 1)) *
Den a p q c s = 0 := by
unfold T at hT
nlinarith [hcenter, hT]
have hlinear :
4 * ((Ο - c) * u - s*v) = Ο^2 - 1 := by
have hz :
4 * ((Ο - c) * u - s*v) - (Ο^2 - 1) = 0 :=
(mul_eq_zero.mp hmul).resolve_right hden
linarith
nlinarith [hlinear, hunit]
/-!
## Geometry bridge
For the coordinate choices in `problem2_solution_en.md`, the three geometric
angle hypotheses yield `Hq = 0`, `Hp = 0`, and the cleared circumcentre
equation `hcenter`. Encoding directed Euclidean angles is independent of the
polynomial certificate above and can be added without changing
`polynomial_certificate` or `algebraic_finish`.
-/
end Problem2
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