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TriangleCenter

[Experimental]

✖
TriangleCenter[tri,type]

gives the specified type of center for the triangle 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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the incenter of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-g2ucl3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-46x1qs
Out[2]=2

Calculate the incenter of a triangle:

In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-dyb72t
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0puv80fwnm-2cuo8o
Out[5]=5

Calculate the excenter of a triangle at the specified vertex:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-k8g0w0
Out[1]=1

Specify a different vertex:

In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-c3yta6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-9bh167
Out[3]=3

Calculate all of the excenters:

In[4]:=4
✖
https://wolfram.com/xid/0puv80fwnm-2u24g5
Out[4]=4

Scope  (12)Survey of the scope of standard use cases

Calculate the endpoint of an angle bisector:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-9dsjwh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-8cd9sn
Out[3]=3

Calculate the centroid of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-5i5tm9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-foyong
Out[2]=2

Calculate the endpoint of a cevian passing through the orthocenter:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-tgs77t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-px0mgs
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-zppa5u
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0puv80fwnm-jxkrmg
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0puv80fwnm-9fclnv
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0puv80fwnm-puhi9s
Out[6]=6

Calculate the circumcenter of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-9ontjj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-ets0mt
Out[2]=2

Calculate the excenter of a triangle at the specified vertex:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-9ms992
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-zuburv
Out[2]=2

Calculate all of the excenters:

In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-bs1qxj
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-g0o63s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-vniu6j
Out[2]=2

Calculate the incenter of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-of5f01
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-z1h79c
Out[2]=2

Calculate the midpoint of a side of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-x7n4yv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-eyme92
Out[2]=2

Calculate the nine-point center of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-vt7b3x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-ln0nt5
Out[2]=2

Calculate the orthocenter of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-qhxaec
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-sb1o1q
Out[2]=2

Calculate the endpoint of a symmedian:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-jps2w0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-6ndjxz
Out[2]=2

Calculate the symmedian point of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-xcq5j3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-b0l0m6
Out[2]=2

Properties & Relations  (20)Properties of the function, and connections to other functions

Angle Bisector and Incenter  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-j8w3cf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-dcm9v2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0puv80fwnm-pmj00z
Out[4]=4

The angle bisectors of a triangle intersect at the incenter:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-omhev1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-bm466n
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-q8z64e
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-5icuh8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-kuoj2p
Out[2]=2

Median, Midpoint and Centroid  (3)

A median intersects the opposite side at the midpoint:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-8yzr3z
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-begiq1
Out[3]=3

The medians of a triangle intersect at the centroid:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-etitue
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-uok388
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-pemnpj
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-khg8nw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-621e1x
Out[2]=2

Perpendicular Bisector, Midpoint and Circumcenter  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-h2ko82
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-jgjq9g
Out[3]=3

The perpendicular bisectors of a triangle intersect at the circumcenter:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-s32aow
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-hik7xg
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-7oiji6
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-7rw40i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-0mjlkq
Out[2]=2

Altitude, Foot and Orthocenter  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-6h1in4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-lqun39
Out[3]=3

The altitudes of a triangle intersect at the orthocenter:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-cun18w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-eo83mi
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-dm6vgv
Out[3]=3

Symmedian, Median and Angle Bisector  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-3557r9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-myh7qc
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-7mbuut
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-0z5hxx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-57jtge
Out[3]=3

The symmedians of a triangle intersect at the symmedian point:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-z4ayvb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-i15lsb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-tipkot
Out[3]=3

Exterior Angle Bisector and Excenter  (2)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-wk8tv7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-tigenk
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0puv80fwnm-8ckx1c
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0puv80fwnm-ijawbu
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-lh74d4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-u3o219
Out[2]=2

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:

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-tz2iev
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-tln5l2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0puv80fwnm-jcwrj9
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0puv80fwnm-pzi0q6
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0puv80fwnm-55cyq9
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0puv80fwnm-2nq410
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0puv80fwnm-kvf0xh
Out[8]=8

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-c0dkwv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-oep5t6
Out[2]=2

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

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-p6ewdv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0puv80fwnm-4hishc
Out[3]=3

Midpoint  (1)

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

In[1]:=1
✖
https://wolfram.com/xid/0puv80fwnm-j55usf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0puv80fwnm-tkzn33
Out[2]=2
Wolfram Research (2019), TriangleCenter, Wolfram Language function, https://reference.wolfram.com/language/ref/TriangleCenter.html.
✖
Wolfram Research (2019), TriangleCenter, Wolfram Language function, https://reference.wolfram.com/language/ref/TriangleCenter.html.

Text

Wolfram Research (2019), TriangleCenter, Wolfram Language function, https://reference.wolfram.com/language/ref/TriangleCenter.html.

✖
Wolfram Research (2019), TriangleCenter, Wolfram Language function, https://reference.wolfram.com/language/ref/TriangleCenter.html.

CMS

Wolfram Language. 2019. "TriangleCenter." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TriangleCenter.html.

✖
Wolfram Language. 2019. "TriangleCenter." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TriangleCenter.html.

APA

Wolfram Language. (2019). TriangleCenter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TriangleCenter.html

✖
Wolfram Language. (2019). TriangleCenter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TriangleCenter.html

BibTeX

@misc{reference.wolfram_2025_trianglecenter, author="Wolfram Research", title="{TriangleCenter}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/TriangleCenter.html}", note=[Accessed: 09-July-2025 ]}

✖
@misc{reference.wolfram_2025_trianglecenter, author="Wolfram Research", title="{TriangleCenter}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/TriangleCenter.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_trianglecenter, organization={Wolfram Research}, title={TriangleCenter}, year={2019}, url={https://reference.wolfram.com/language/ref/TriangleCenter.html}, note=[Accessed: 09-July-2025 ]}

✖
@online{reference.wolfram_2025_trianglecenter, organization={Wolfram Research}, title={TriangleCenter}, year={2019}, url={https://reference.wolfram.com/language/ref/TriangleCenter.html}, note=[Accessed: 09-July-2025 ]}