Formal evidence
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Geometry · formal theorem
Heron’s Formula — Coordinate Identity
For any three points in ℝ², the square of twice their signed coordinate area equals four times the Heron radicand of their side lengths.
Lean proofpassed
Unfinished proof stepsNone
Formal resultAccepted
Line counts exclude blank lines; comments and documentation count. The total is the commit-pinned first-party Lean import closure; Mathlib and other third-party dependencies are excluded.
The theorem at a glance
Heron's coordinate identity at a glance
Accessible transcript
(2Aₛ)² = 4s(s − a)(s − b)(s − c)
The checked proof expands Heron's four factors, rewrites them through squared side lengths, and matches the resulting polynomial with the squared coordinate determinant. It establishes the squared twice-signed-area identity for any three planar points, not a separate unsigned square-root formula.
Read the complete poster transcript
PLANE GEOMETRY · COORDINATE IDENTITY
HERON’S FORMULA
COORDINATES AND SIDE LENGTHS MEASURE THE SAME AREA
FOR ANY p, q, r ∈ ℝ²
a = |q − r| · b = |r − p| · c = |p − q|
s = (a + b + c)/2
(2Aₛ)² = 4s(s − a)(s − b)(s − c)
Aₛ is the signed coordinate area.
Squaring removes the choice of orientation.
r = (x₃, y₃)
b = |r − p| · a = |q − r|
2Aₛ = |x₂ − x₁ y₂ − y₁; x₃ − x₁ y₃ − y₁|
O · p = (x₁, y₁) · c = |p − q| · q = (x₂, y₂)
HOW THE PROOF MOVES
1 · EXPAND HERON’S FOUR FACTORS
16s(s − a)(s − b)(s − c)
2 · REWRITE USING SIDE SQUARES
4a²b² − (a² + b² − c²)²
3 · USE THE COORDINATE DETERMINANT
4a²b² − (a² + b² − c²)² = 4(2Aₛ)²
4 · COMPARE THE IDENTITIES
(2Aₛ)² = 4s(s − a)(s − b)(s − c)
EXACT SCOPE
The squared twice-signed-area identity for three points in ℝ².
No nondegeneracy assumption. Not a separate proof of an unsigned square-root formula.
Theorem schematic
One area in two mathematical languages
(2Aₛ)² = 4s(s − a)(s − b)(s − c)
For any three points in ℝ², the square of twice their signed coordinate area equals four times s(s − a)(s − b)(s − c), where a, b, c are the side lengths and s is their semiperimeter. Squaring removes orientation and permits degenerate triples.
Exact formal proposition
Hypotheses and conclusion
This is the meaningful proposition proved by the checked wrapper declaration. It is extracted from the same commit-pinned Lean source.
noncomputable def HeronFormulaStatement : Prop :=
∀ p q r : Point,
(twiceSignedArea p q r) ^ 2 =
4 * heronRadicand (side q r) (side r p) (side p q)Definitions used in this proposition
Point
abbrev Point : Type :=
Fin 2 → ℝsqDist
def sqDist (p q : Point) : ℝ :=
(p 0 - q 0) ^ 2 + (p 1 - q 1) ^ 2side
noncomputable def side (p q : Point) : ℝ :=
Real.sqrt (sqDist p q)twiceSignedArea
def twiceSignedArea (p q r : Point) : ℝ :=
(q 0 - p 0) * (r 1 - p 1) - (q 1 - p 1) * (r 0 - p 0)heronRadicand
noncomputable def heronRadicand (a b c : ℝ) : ℝ :=
let s := (a + b + c) / 2
s * (s - a) * (s - b) * (s - c)Lean wrapper declaration
The checked endpoint names the proposition displayed above. This short declaration is useful for source identity, but the expanded proposition is the mathematical statement to read first.
theorem heronFormula :
HeronFormulaStatementResult boundary
What this page does—and does not—establish
For any three points in ℝ², including collinear triples, the square of twice their signed coordinate area equals four times the Heron radicand of their side lengths.
- Because the identity is squared, it does not select an orientation or separately state a positive unsigned-area formula.
About these visual explanations
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Formal-result publication and review details
Independent publication review
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The formal declaration was accepted against the named theorem and its exact variant.
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Accepted result
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