Fermat point
E146192
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Torricelli point | 2 |
| Fermat point canonical | 1 |
| Fermat–Torricelli point | 1 |
| Fermat’s minimal path problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1281487 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fermat point Context triple: [Pierre de Fermat, notableWork, Fermat point]
-
A.
The Angle
The Angle is a prominent battlefield site at Gettysburg, marking the focal point of Pickett’s Charge and often referred to as the “High Water Mark of the Confederacy.”
-
B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
C.
Peuterey Integral
Peuterey Integral is a legendary, highly committing alpine climbing route that follows the full Peuterey ridge to the summit of Mont Blanc, renowned as one of the longest and most serious ridge climbs in the Alps.
-
D.
Géométrie de position
Géométrie de position is a foundational 1803 treatise by Lazare Carnot that helped establish projective geometry and modern geometric reasoning about position and transformation.
-
E.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fermat point Target entity description: The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
A.
The Angle
The Angle is a prominent battlefield site at Gettysburg, marking the focal point of Pickett’s Charge and often referred to as the “High Water Mark of the Confederacy.”
-
B.
Cauchy-à-la-Tour
Cauchy-à-la-Tour is a small commune in the Pas-de-Calais department of northern France.
-
C.
Peuterey Integral
Peuterey Integral is a legendary, highly committing alpine climbing route that follows the full Peuterey ridge to the summit of Mont Blanc, renowned as one of the longest and most serious ridge climbs in the Alps.
-
D.
Géométrie de position
Géométrie de position is a foundational 1803 treatise by Lazare Carnot that helped establish projective geometry and modern geometric reasoning about position and transformation.
-
E.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle of geometry stating that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
geometric point
ⓘ
special point of a triangle ⓘ triangle center ⓘ |
| alsoKnownAs |
Fermat–Torricelli point
ⓘ
linked to:
Fermat point
Torricelli point ⓘ
linked to:
Fermat point
|
| angleCondition | construction with equilateral triangles valid when all angles of the triangle are less than 120 degrees ⓘ |
| angleProperty | forms 120-degree angles between segments to the vertices in an acute triangle ⓘ |
| appliesTo | triangle ⓘ |
| barycentricCoordinates | a·csc(A+π/3) : b·csc(B+π/3) : c·csc(C+π/3) ⓘ |
| belongsTo | set of triangle centers catalogued in ETC ⓘ |
| category |
optimization in geometry
ⓘ
triangle geometry ⓘ |
| constructionMethod |
intersection of lines from vertices to outer vertices of equilateral triangles constructed externally on each side
ⓘ
intersection of three lines each making 120 degrees with two sides of the triangle ⓘ |
| coordinateSystem | has known trilinear coordinates ⓘ |
| definedIn | Euclidean geometry ⓘ |
| differenceFrom | geometric median for more than three points ⓘ |
| distanceProperty | sum of distances from Fermat point to vertices is minimal among all points in the plane ⓘ |
| ETCIndex | X(13) ⓘ |
| existenceCondition | unique for any nondegenerate triangle ⓘ |
| generalizationOf | median point for three terminals in the plane ⓘ |
| geometricProperty | is the unique point where the three segments to vertices meet at 120 degrees in an acute triangle ⓘ |
| historicalNote | problem posed by Pierre de Fermat in the 17th century ⓘ |
| invariantUnder |
rigid motions of the plane
ⓘ
similarity transformations of the triangle ⓘ |
| liesOn | lines joining each vertex to the opposite equilateral triangle vertex in the classical construction ⓘ |
| locationProperty |
lies inside an acute triangle
ⓘ
lies on a vertex of an obtuse triangle ⓘ |
| namedAfter |
Evangelista Torricelli
ⓘ
Pierre de Fermat ⓘ |
| optimizationProperty |
gives minimal network length connecting the three vertices with one Steiner point
ⓘ
minimizes sum of distances to the three vertices of the triangle ⓘ |
| relatedConcept |
Fermat problem
ⓘ
linked to:
Fermat's Last Theorem
Steiner tree problem ⓘ geometric median of three points ⓘ |
| relatedTo |
Fermat’s minimal path problem
ⓘ
linked to:
Fermat point
|
| relatedTriangleCenter |
centroid
ⓘ
circumcenter ⓘ incenter ⓘ orthocenter ⓘ |
| solvedBy | Evangelista Torricelli ⓘ |
| specialCase | coincides with the obtuse vertex when the triangle has an angle of at least 120 degrees ⓘ |
| symmetryProperty | symmetric with respect to permutations of the triangle’s vertices ⓘ |
| trilinearCoordinates | csc(A+π/3) : csc(B+π/3) : csc(C+π/3) ⓘ |
| usedIn |
facility location problems
ⓘ
geometric optimization ⓘ network optimization ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fermat point Description of subject: The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Fermat point
linked to: Fermat point
linked to: Fermat point
linked to: Fermat point