Miquel circle
E913508
The Miquel circle is a notable circle in geometry that passes through the three points where the circumcircles of the triangles formed by choosing three vertices of a quadrilateral intersect.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Miquel circle canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11242695 — 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 circle Context triple: [Conway circle theorem, relatedTo, Miquel circle]
-
A.
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.
-
B.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
-
C.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
D.
Malfatti
Malfatti is an Italian-origin surname notably associated with Brazilian modernist painter Anita Malfatti.
-
E.
Tusi couple
The Tusi couple is a geometric device from medieval Islamic astronomy that generates linear motion from the sum of two circular motions, later influencing Copernican models of planetary motion.
- 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 circle Target entity description: The Miquel circle is a notable circle in geometry that passes through the three points where the circumcircles of the triangles formed by choosing three vertices of a quadrilateral intersect.
-
A.
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.
-
B.
Conway circle theorem
The Conway circle theorem is a geometric result in triangle geometry that identifies a special circle associated with a triangle and certain constructed points, revealing notable collinearities and concyclicity relationships.
-
C.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
D.
Malfatti
Malfatti is an Italian-origin surname notably associated with Brazilian modernist painter Anita Malfatti.
-
E.
Tusi couple
The Tusi couple is a geometric device from medieval Islamic astronomy that generates linear motion from the sum of two circular motions, later influencing Copernican models of planetary motion.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
circle in Euclidean geometry
ⓘ
geometric concept ⓘ |
| appearsIn |
literature on Miquel configurations
ⓘ
treatises on triangle and circle geometry ⓘ |
| appliesTo |
complete quadrilateral
ⓘ
quadrilateral ⓘ |
| classification |
circle associated with a complete quadrilateral
ⓘ
circle associated with a quadrilateral ⓘ |
| configurationElement |
Miquel point
NERFINISHED
ⓘ
circumcircles of the three triangles ⓘ four vertices of a quadrilateral ⓘ three pairwise intersection points of the circumcircles ⓘ three triangles formed by choosing three of the four vertices ⓘ |
| constructionUses | circumcircles of triangles formed by three vertices of a quadrilateral ⓘ |
| definedFor |
four points in the plane
ⓘ
quadrilateral with four vertices ⓘ |
| dependsOn |
existence of circumcircles of the three triangles
ⓘ
non-collinearity of the quadrilateral vertices ⓘ |
| field |
Euclidean geometry
NERFINISHED
ⓘ
geometry ⓘ |
| generalizationOf | Miquel configuration for polygons NERFINISHED ⓘ |
| geometricNature | locus of points concyclic with the three circumcircle intersection points ⓘ |
| hasCenter | center determined uniquely by the three intersection points ⓘ |
| hasPoint | each of the three pairwise intersection points of the circumcircles ⓘ |
| hasRadius | radius determined by distance from its center to any of the three intersection points ⓘ |
| hasType | circle determined by intersection points of circumcircles ⓘ |
| namedAfter | Auguste Miquel NERFINISHED ⓘ |
| namedForRole | Auguste Miquel’s work on circle configurations ⓘ |
| occursIn |
classical Euclidean geometry
ⓘ
elementary geometry ⓘ |
| passesThrough |
Miquel point of the quadrilateral
NERFINISHED
ⓘ
three pairwise intersection points of circumcircles of triangles from a quadrilateral ⓘ |
| property |
the three circumcircles of triangles formed from a quadrilateral concur in a single point (Miquel point)
ⓘ
the three intersection points of the circumcircles are concyclic ⓘ unique for a given quadrilateral configuration ⓘ |
| relatedTo |
Miquel point
NERFINISHED
ⓘ
Miquel theorem NERFINISHED ⓘ circumcircle ⓘ complete quadrilateral ⓘ cyclic quadrilateral ⓘ |
| symmetryProperty | invariant under permutations of the four vertices of the quadrilateral ⓘ |
| theoremInvolves | Miquel theorem for quadrilaterals NERFINISHED ⓘ |
| usedIn |
Olympiad geometry
ⓘ
geometric problem solving ⓘ synthetic geometry proofs ⓘ |
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 circle Description of subject: The Miquel circle is a notable circle in geometry that passes through the three points where the circumcircles of the triangles formed by choosing three vertices of a quadrilateral intersect.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.