In the checked nondegenerate butterfly configuration, the opposite-chord intersections X and Y have the original chord midpoint M as their midpoint.
Scope: For the recorded nondegenerate coordinate-circle butterfly configuration, M is the midpoint of the two uniquely specified opposite-chord intersections X and Y.
Lean checkedBuild passed
Unfinished proof stepsNone
PublicationAccepted formal theorem
Starting pointCircle, chord midpoint M, two chords through M
→
RelationOpposite-chord intersections
→
ConclusionM is the midpoint of X and Y
Exact scope: For the recorded nondegenerate coordinate-circle butterfly configuration, M is the midpoint of the two uniquely specified opposite-chord intersections X and Y.
Follow the coordinate proof from the exact nondegenerate configuration through normalization, complete chart coverage, secant relations, and transport back.
These stages follow the checked source and explain the mathematical route. They summarize the argument; they are not a visualization of Lean's internal proof term.
The single base carries X, P, M, Q, and Y; through-M chords AB and CD are crossed by the opposite-endpoint lines AD and BC.
01
Fix the exact nondegenerate configuration
The unit circle uses the base y=0.3 with P=(-√0.91,0.3), M=(0,0.3), and Q=(√0.91,0.3). The chord AB lies on y=0.7x+0.3 and the chord CD lies on y=-1.2x+0.3. Their opposite-endpoint lines AD and BC meet the same base at exterior points X and Y. The nine marked points and five complete lines encode the retained nondegenerate hypotheses without asserting that all six circle endpoints are pairwise distinct.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
The arbitrary configuration is transported to a normalized base coordinate.
02
Normalize the base chord
baseNormalize
An affine coordinate change sends the tilted base chord to a horizontal normal form while preserving the relevant line incidences and midpoint relation.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
Finite and vertical coordinate charts cover both auxiliary chords.
03
Cover the four auxiliary-chord charts
Each of the two through-M chords is either represented by a finite slope or a vertical chart, producing four cases that cover the normalized configuration.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
Secant-pair relations constrain the normalized endpoints.
04
Apply the circle secant relations
Circle membership links the endpoint coordinate pairs on each through-M chord, supplying the algebraic relations used by the intersection formulas.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
The two base intersections sit at opposite coordinates around the normalized midpoint.
05
Derive opposite base coordinates
x_X = -x_Y
Solving the two unique line intersections shows that their normalized base coordinates are negatives of one another.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
Every coordinate chart proves the same normalized midpoint equality.
06
All charts return the same midpoint
0 = midpoint(x_X,x_Y)
The finite/vertical case split converges on one statement: the normalized origin is the midpoint of the two base intersections.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
The midpoint result returns from normalized coordinates to the original geometry.
07
Transport the midpoint back
M = midpoint(X,Y)
Undoing the normalization preserves midpoint and incidence, carrying the normalized symmetry back to the original tilted circle configuration.
Lean lemmas for this step
ButterflyConfiguration
ButterflyHyp
DistinctChordEndpoints
UniqueIntersection
baseNormalize
baseNormalizedCheckedChartCoverage
baseNormalizedButterflyTheorem
nondegenerateButterflyTheorem
The theorem at a glance
Butterfly Theorem at a glance
A chord midpoint forces the two crossed-line intersections to be symmetric about it.
Accessible transcript
M = midpoint(X,Y)
The exact nondegenerate midpoint statement and coordinate proof route are presented without risking a misleading labeled construction.
Read the complete poster transcript
EUCLIDEAN GEOMETRY · CIRCLE AND CHORDS
NONDEGENERATE BUTTERFLY THEOREM
ONE CHORD MIDPOINT FORCES ANOTHER
M = midpoint(P,Q)
X = PQ ∩ AD
Y = PQ ∩ BC
M = midpoint(X,Y)
P ≠ Q · A ≠ B · C ≠ D
Both base intersections are unique.
HOW THE PROOF MOVES
NORMALIZE THE BASE CHORD · COVER FINITE AND VERTICAL CHARTS · APPLY THE SECANT RELATIONS · DERIVE OPPOSITE BASE COORDINATES · TRANSPORT THE MIDPOINT BACK
EXACT SCOPE
Distinct chord endpoints in the three stated pairs and unique intersections; collapsed same-chord configurations are excluded.
Theorem schematic
A midpoint symmetry hidden in crossed chords
Two asymmetric through-midpoint chords generate exterior base intersections equidistant from the original midpoint.
M = midpoint(X,Y)
The base chord lies above the circle center and contains M. Two complete chords of different slopes pass through M; their opposite endpoints determine two complete wing lines whose exterior intersections X and Y lie on the base with XM=MY.
Result boundary
What this theorem does—and does not—establish
For the recorded nondegenerate coordinate-circle butterfly configuration, M is the midpoint of the two uniquely specified opposite-chord intersections X and Y.
The superseded v0 statement is false because it omits necessary nondegeneracy and uniqueness conditions; only the repaired endpoint is claimed.
About these visual explanations
These AI-generated visuals explain the theorem and proof route; they are not proof evidence. Their publication review was completed separately from review of the formal result. The exact Lean proposition and checked source remain authoritative.
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.
Continue the mathematics
Open questions and extensions
The pinned theorem and its complete local import closure let people and AI agents inspect the proof, compare another route, isolate reusable lemmas, or formulate a stronger exact statement. Lean checks each proposed extension against its own exact statement.
What the source ZIP contains
The ZIP contains the checked first-party Lean import closure, exact statements and boundaries, license, notice, evidence, source-footprint manifest, and continuation data. Mathlib and other third-party dependencies are not bundled.
Lean checks the exact proof. Accepted review records cover evidence completeness, statement alignment, scope, and public wording; the visual review covers explanation only.
01
Formal evidence
The formal-evidence review accepted the recorded build, exact declarations, unfinished-step scan, and axiom evidence.
02
Statement alignment
The formal declaration was accepted against the named theorem and its exact variant.
03
Result boundary
The accepted boundary keeps nearby stronger or commonly confused claims out of scope.
04
Public wording
The public-wording review accepted the retained theorem explanation and source presentation. Generated media follows a separate review and promotion gate.
05
Canonical source
The first-party source link is pinned to the checked package commit and exact Lean file.
06
Accepted result
A validated accepted-result record binds the four reviews to the checked formalization.
