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.
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
geometric point
ⓘ
triangle center ⓘ |
| alsoKnownAs | Miquel point of a triangle NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Miquel configuration NERFINISHED ⓘ Miquel theorem NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.