TriangleCenter

TriangleCenter[tri,type]

gives the specified type of center for the triangle tri.

TriangleCenter[tri]

gives the centroid of the triangle.

Details

  • TriangleCenter gives a list of coordinates.
  • The triangle tri can be given as {p1,p2,p3}, Triangle[{p1,p2,p3}] or Polygon[{p1,p2,p3}].
  • The following center types can be given:
  • {"AngleBisectingCevianEndpoint",p}endpoint of the cevian bisecting the angle at the vertex p
    "Centroid"centroid
    {"CevianEndpoint",center,p}endpoint of the cevian passing through the vertex p and the specified center
    "Circumcenter"center of the circumcircle
    {"Excenter",p}center of the excircle opposite from the vertex p
    {"Foot",p}foot of the altitude passing through the vertex p
    "Incenter"center of the incircle
    {"Midpoint",p}midpoint of the side opposite from the vertex p
    "NinePointCenter"center of nine-point circle
    "Orthocenter"orthocenter
    {"SymmedianEndpoint",p}endpoint of the symmedian passing through the vertex p
    "SymmedianPoint"symmedian point
  • In the form {"type",p}, p can be a symbolic point specification in a GeometricScene, or it can be an explicit vertex of the form {x,y}, Point[{x,y}] or the index i of the vertex. When given in the short form "type", the vertex p2 is used.
  • In the form {"CevianEndpoint",center,p}, the center can be given as a center type such as "Centroid" or as a point specification. When given in the short form {"CevianEndpoint",center}, the vertex p2 is used.
  • In any form that specifies a vertex p, a value of All will return a list of three values corresponding to the vertices.
  • TriangleCenter can be used with symbolic points in GeometricScene.

Examples

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Basic Examples  (2)

Find the incenter of a triangle:

Calculate the incenter of a triangle:

Calculate the excenter of a triangle at the specified vertex:

Specify a different vertex:

Calculate all of the excenters:

Scope  (12)

Calculate the endpoint of an angle bisector:

Calculate the centroid of a triangle:

Calculate the endpoint of a cevian passing through the orthocenter:

Calculate the endpoint of a cevian passing through a different vertex:

Calculate the endpoint of a cevian through an arbitrary center point:

Calculate the circumcenter of a triangle:

Calculate the excenter of a triangle at the specified vertex:

Calculate all of the excenters:

Calculate the foot of an altitude of a triangle at the specified vertex:

Calculate the incenter of a triangle:

Calculate the midpoint of a side of a triangle:

Calculate the nine-point center of a triangle:

Calculate the orthocenter of a triangle:

Calculate the endpoint of a symmedian:

Calculate the symmedian point of a triangle:

Properties & Relations  (20)

Angle Bisector and Incenter  (3)

An angle bisector endpoint is the intersection of an angle bisector and the opposite side:

The angle bisectors of a triangle intersect at the incenter:

TriangleConstruct[{a,b,c},"Incircle"] is equivalent to Circle[TriangleCenter[{a,b,c},"Incenter"],TriangleMeasurement[{a,b,c},"Inradius"]]:

Median, Midpoint and Centroid  (3)

A median intersects the opposite side at the midpoint:

The medians of a triangle intersect at the centroid:

TriangleCenter[{a,b,c},"Centroid"] is equivalent to RegionCentroid[Triangle[{a,b,c}]]:

Perpendicular Bisector, Midpoint and Circumcenter  (3)

The perpendicular bisector of a side passes through the midpoint of that side:

The perpendicular bisectors of a triangle intersect at the circumcenter:

TriangleConstruct[{a,b,c},"Circumcircle"] is equivalent to Circle[TriangleCenter[{a,b,c},"Circumcenter"],TriangleMeasurement[{a,b,c},"Circumradius"]]:

Altitude, Foot and Orthocenter  (2)

The foot of an altitude is the intersection of the altitude and the opposite side:

The altitudes of a triangle intersect at the orthocenter:

Symmedian, Median and Angle Bisector  (3)

The endpoint of a symmedian at a vertex is the intersection of the symmedian and the opposite side:

The angle bisector at a vertex also bisects the angle formed by the median and symmedian at that vertex:

The symmedians of a triangle intersect at the symmedian point:

Exterior Angle Bisector and Excenter  (2)

The excenter opposite a vertex is the intersection of the exterior angle bisectors of the opposite angles:

TriangleConstruct[{a,b,c},"Excircle"] is equivalent to Circle[TriangleCenter[{a,b,c},"Excenter"],TriangleMeasurement[{a,b,c},"Exradius"]]:

Nine-Point Circle, Foot, Midpoint, Orthocenter  (2)

The nine-point circle of a triangle passes through the feet of the altitudes, the midpoints of the sides and the midpoints of the segments from the vertices to the orthocenter:

TriangleConstruct[{a,b,c},"NinePointCircle"] is equivalent to Circle[TriangleCenter[{a,b,c},"NinePointCenter"],TriangleMeasurement[{a,b,c},"NinePointRadius"]]:

Euler Line, Centroid, Circumcenter, Orthocenter and Nine-Point Center  (1)

The Euler line passes through the centroid, circumcenter, orthocenter and nine-point center:

Midpoint  (1)

TriangleCenter[{a,b,c},"Midpoint"] is equivalent to Midpoint[{a,c}]:

Introduced in 2019
 (12.0)