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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.