Miquel point
E913507
The Miquel point is a notable point in triangle geometry defined as the common intersection of the circumcircles of three triangles formed by choosing one point on each side of a reference triangle.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Miquel point canonical | 3 |
| Miquel point of a triangle | 1 |
| Miquel point of the quadrilateral | 1 |
| Miquel theorem | 1 |
| Steiner point in triangle geometry | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11242694 — 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: Miquel point Context triple: [Conway circle theorem, relatedTo, Miquel point]
-
A.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
B.
Hattenbacher Dreieck
Hattenbacher Dreieck is a major German motorway interchange in Hesse where key Autobahn routes converge, serving as an important junction in the national highway network.
-
C.
Soddy circle
A Soddy circle is one of the circles in a configuration of four mutually tangent circles, central to the geometric problem described by Descartes' circle theorem.
-
D.
Kiepert
Kiepert is a character in Bertolt Brecht’s early play "Drums in the Night," which explores post–World War I German society and disillusionment.
-
E.
Malfatti
Malfatti is an Italian-origin surname notably associated with Brazilian modernist painter Anita Malfatti.
- 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: Miquel point Target entity description: The Miquel point is a notable point in triangle geometry defined as the common intersection of the circumcircles of three triangles formed by choosing one point on each side of a reference triangle.
-
A.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
B.
Hattenbacher Dreieck
Hattenbacher Dreieck is a major German motorway interchange in Hesse where key Autobahn routes converge, serving as an important junction in the national highway network.
-
C.
Soddy circle
A Soddy circle is one of the circles in a configuration of four mutually tangent circles, central to the geometric problem described by Descartes' circle theorem.
-
D.
Kiepert
Kiepert is a character in Bertolt Brecht’s early play "Drums in the Night," which explores post–World War I German society and disillusionment.
-
E.
Malfatti
Malfatti is an Italian-origin surname notably associated with Brazilian modernist painter Anita Malfatti.
- F. None of above. chosen
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
geometric point
ⓘ
triangle center ⓘ |
| alsoKnownAs |
Miquel point of a triangle
ⓘ
linked to:
Miquel point
|
| appearsIn | classical Euclidean geometry literature ⓘ |
| associatedWith |
complete quadrilateral
ⓘ
spiral similarity ⓘ |
| belongsTo | set of notable points of a triangle ⓘ |
| category | triangle center depending on three points on sides ⓘ |
| constructionMethod | intersection of circumcircles of three triangles formed by choosing one point on each side of a reference triangle ⓘ |
| constructionUses |
circumcircle
ⓘ
points on sides of triangle ⓘ |
| coordinateSystems |
can be expressed in barycentric coordinates
ⓘ
can be expressed in trilinear coordinates ⓘ |
| definedFor | triangle ⓘ |
| dependsOn | reference triangle ⓘ |
| field |
Euclidean geometry
ⓘ
triangle geometry ⓘ |
| invariantUnder |
choice of points on sides (one per side)
ⓘ
cyclic permutation of triangle vertices ⓘ |
| liesOn |
circumcircle of triangle formed by two vertices of reference triangle and chosen point on another side
ⓘ
circumcircle of triangle formed by two vertices of reference triangle and chosen point on included side ⓘ circumcircle of triangle formed by two vertices of reference triangle and chosen point on the third side ⓘ |
| namedAfter | Auguste Miquel NERFINISHED ⓘ |
| namedEntity | yes ⓘ |
| property | independent of specific choice of points on each side as long as one point is chosen on each side ⓘ |
| relatedConcept |
Miquel circle
ⓘ
Miquel configuration ⓘ
linked to:
Miquel circle
Miquel theorem ⓘ
linked to:
Miquel point
|
| role | center of spiral similarity between pairs of segments determined by triangle vertices and chosen points ⓘ |
| theoremStates | for any triangle and any choice of one point on each side, the three circumcircles of the triangles formed by the vertices and these points are concurrent at the Miquel point ⓘ |
| usedIn |
configurations involving concurrent circumcircles
ⓘ
synthetic proofs in triangle geometry ⓘ |
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: Miquel point Description of subject: The Miquel point is a notable point in triangle geometry defined as the common intersection of the circumcircles of three triangles formed by choosing one point on each side of a reference triangle.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Miquel point
linked to: Miquel point
linked to: Miquel point
linked to: Miquel point