Difference between revisions of "Center (geometry)"
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==Triangle centers== | ==Triangle centers== | ||
| − | *The [[centroid]] is where the three [[median]]s of the triangle meet. | + | *The [[centroid]] <math>G</math> is where the three [[median]]s of the triangle meet. |
| − | *The [[incenter]] of the triangle is where the three [[angle bisector]]s meet. It is also the center of the [[incircle]]. | + | *The [[incenter]] <math>I</math> of the triangle is where the three [[angle bisector]]s meet. It is also the center of the [[incircle]]. |
| − | *The [[circumcenter]] is where the [[perpendicular bisector]]s of the triangles sides meet. It is also the center of the [[circumcircle]]. | + | *The [[circumcenter]] <math>O</math> is where the [[perpendicular bisector]]s of the triangles sides meet. It is also the center of the [[circumcircle]]. |
| − | *The [[orthocenter]] Is where the [[altitude]]s of the triangle meet. | + | *The [[orthocenter]] <math>H</math> Is where the [[altitude]]s of the triangle meet. |
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| + | Other notable centers include the nine-point center, symmedian point (Lemoine point), Nagel point, and Gergonne point. | ||
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| + | Facts: | ||
| + | The centroid, circumcenter, orthocenter, and nine-point center are always colinear on the Euler line. | ||
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| + | The incenter of a triangle is the orthocenter of its excentral triangle, and the circumcenter of a triangle is the nine-point center of its excentral triangle, so the line connecting the circumcenter and incenter of a triangle is the Euler line of its excentral triangle. | ||
{{stub}} | {{stub}} | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Latest revision as of 21:55, 7 October 2025
The center of a circle or sphere is a point inside the circle which is equidistant from all points on the circle.
Triangle centers
is where the three - The incenter
of the triangle is where the three angle bisectors meet. It is also the center of the incircle.
of the triangle is where the three - The circumcenter
is where the perpendicular bisectors of the triangles sides meet. It is also the center of the circumcircle.
is where the - The orthocenter
Is where the altitudes of the triangle meet.
Is where the Other notable centers include the nine-point center, symmedian point (Lemoine point), Nagel point, and Gergonne point.
Facts:
The centroid, circumcenter, orthocenter, and nine-point center are always colinear on the Euler line.
The incenter of a triangle is the orthocenter of its excentral triangle, and the circumcenter of a triangle is the nine-point center of its excentral triangle, so the line connecting the circumcenter and incenter of a triangle is the Euler line of its excentral triangle.
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