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.
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
circle in triangle geometry
ⓘ
geometric object ⓘ |
| alsoKnownAs |
Euler circle
NERFINISHED
ⓘ
Feuerbach circle NERFINISHED ⓘ nine-point conic ⓘ |
| centerLiesOn | Euler line NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
centroid of the triangle ⓘ circumcenter of the triangle ⓘ orthocenter of the triangle ⓘ |
| studiedIn | Euclidean geometry NERFINISHED ⓘ |
| tangencyPointWithIncircle | Feuerbach point NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.