nine-point circle
E913509
The nine-point circle is a fundamental circle in triangle geometry that passes through nine significant points of a triangle, including the midpoints of its sides and the feet of its altitudes.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Euler circle | 1 |
| Feuerbach circle | 1 |
| nine-point circle canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11242696 — 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: nine-point circle Context triple: [Conway circle theorem, relatedTo, nine-point 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.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
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C.
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.
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D.
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.
-
E.
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.
- 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: nine-point circle Target entity description: The nine-point circle is a fundamental circle in triangle geometry that passes through nine significant points of a triangle, including the midpoints of its sides and the feet of its altitudes.
-
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.
Fermat point
The Fermat point is a special point inside a triangle that minimizes the total distance to the triangle’s three vertices.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
circle in triangle geometry
ⓘ
geometric object ⓘ |
| alsoKnownAs |
Euler circle
ⓘ
linked to:
nine-point circle
Feuerbach circle ⓘ
linked to:
nine-point circle
nine-point conic ⓘ |
| centerLiesOn | Euler line ⓘ |
| contains |
feet of the three altitudes of the triangle
ⓘ
midpoints of segments joining orthocenter to the three vertices ⓘ midpoints of the three sides of the triangle ⓘ |
| definedFor | triangle ⓘ |
| existsFor | every nondegenerate triangle ⓘ |
| hasCenter | nine-point center ⓘ |
| hasSymmetry | invariance under permutations of triangle vertices ⓘ |
| homothetyCenter | orthocenter of the triangle ⓘ |
| homothetyRatio | 1/2 ⓘ |
| isHomotheticTo | circumcircle of the triangle ⓘ |
| liesInside | circumcircle of the triangle ⓘ |
| namedAfter | Karl Wilhelm Feuerbach ⓘ |
| passesThrough |
foot of altitude from vertex A
ⓘ
foot of altitude from vertex B ⓘ foot of altitude from vertex C ⓘ midpoint of segment from orthocenter to vertex A ⓘ midpoint of segment from orthocenter to vertex B ⓘ midpoint of segment from orthocenter to vertex C ⓘ midpoint of side AB ⓘ midpoint of side BC ⓘ midpoint of side CA ⓘ |
| property | concyclicity of the nine special points ⓘ |
| radiusRelation | radius equals half the circumradius in an acute triangle ⓘ |
| relatedTo |
Euler line
ⓘ
centroid of the triangle ⓘ circumcenter of the triangle ⓘ orthocenter of the triangle ⓘ |
| studiedIn | Euclidean geometry ⓘ |
| tangencyPointWithIncircle | Feuerbach point ⓘ |
| tangentTo |
A-excircle of the triangle
ⓘ
B-excircle of the triangle ⓘ C-excircle of the triangle ⓘ incircle of the triangle ⓘ |
| usedIn |
classical Euclidean constructions
ⓘ
triangle center studies ⓘ |
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: nine-point circle Description of subject: The nine-point circle is a fundamental circle in triangle geometry that passes through nine significant points of a triangle, including the midpoints of its sides and the feet of its altitudes.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: nine-point circle
linked to: nine-point circle