TriangleCenter
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 allBasic Examples (2)Summary of the most common use cases
Find the incenter of a triangle:

https://wolfram.com/xid/0puv80fwnm-g2ucl3


https://wolfram.com/xid/0puv80fwnm-46x1qs

Calculate the incenter of a triangle:

https://wolfram.com/xid/0puv80fwnm-dyb72t


https://wolfram.com/xid/0puv80fwnm-2cuo8o

Calculate the excenter of a triangle at the specified vertex:

https://wolfram.com/xid/0puv80fwnm-k8g0w0


https://wolfram.com/xid/0puv80fwnm-c3yta6


https://wolfram.com/xid/0puv80fwnm-9bh167

Calculate all of the excenters:

https://wolfram.com/xid/0puv80fwnm-2u24g5

Scope (12)Survey of the scope of standard use cases
Calculate the endpoint of an angle bisector:

https://wolfram.com/xid/0puv80fwnm-9dsjwh


https://wolfram.com/xid/0puv80fwnm-8cd9sn

Calculate the centroid of a triangle:

https://wolfram.com/xid/0puv80fwnm-5i5tm9


https://wolfram.com/xid/0puv80fwnm-foyong

Calculate the endpoint of a cevian passing through the orthocenter:

https://wolfram.com/xid/0puv80fwnm-tgs77t


https://wolfram.com/xid/0puv80fwnm-px0mgs

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

https://wolfram.com/xid/0puv80fwnm-zppa5u


https://wolfram.com/xid/0puv80fwnm-jxkrmg

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

https://wolfram.com/xid/0puv80fwnm-9fclnv


https://wolfram.com/xid/0puv80fwnm-puhi9s

Calculate the circumcenter of a triangle:

https://wolfram.com/xid/0puv80fwnm-9ontjj


https://wolfram.com/xid/0puv80fwnm-ets0mt

Calculate the excenter of a triangle at the specified vertex:

https://wolfram.com/xid/0puv80fwnm-9ms992


https://wolfram.com/xid/0puv80fwnm-zuburv

Calculate all of the excenters:

https://wolfram.com/xid/0puv80fwnm-bs1qxj

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

https://wolfram.com/xid/0puv80fwnm-g0o63s


https://wolfram.com/xid/0puv80fwnm-vniu6j

Calculate the incenter of a triangle:

https://wolfram.com/xid/0puv80fwnm-of5f01


https://wolfram.com/xid/0puv80fwnm-z1h79c

Calculate the midpoint of a side of a triangle:

https://wolfram.com/xid/0puv80fwnm-x7n4yv


https://wolfram.com/xid/0puv80fwnm-eyme92

Calculate the nine-point center of a triangle:

https://wolfram.com/xid/0puv80fwnm-vt7b3x


https://wolfram.com/xid/0puv80fwnm-ln0nt5

Calculate the orthocenter of a triangle:

https://wolfram.com/xid/0puv80fwnm-qhxaec


https://wolfram.com/xid/0puv80fwnm-sb1o1q

Calculate the endpoint of a symmedian:

https://wolfram.com/xid/0puv80fwnm-jps2w0


https://wolfram.com/xid/0puv80fwnm-6ndjxz

Calculate the symmedian point of a triangle:

https://wolfram.com/xid/0puv80fwnm-xcq5j3


https://wolfram.com/xid/0puv80fwnm-b0l0m6

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:

https://wolfram.com/xid/0puv80fwnm-j8w3cf


https://wolfram.com/xid/0puv80fwnm-dcm9v2


https://wolfram.com/xid/0puv80fwnm-pmj00z

The angle bisectors of a triangle intersect at the incenter:

https://wolfram.com/xid/0puv80fwnm-omhev1


https://wolfram.com/xid/0puv80fwnm-bm466n


https://wolfram.com/xid/0puv80fwnm-q8z64e

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

https://wolfram.com/xid/0puv80fwnm-5icuh8


https://wolfram.com/xid/0puv80fwnm-kuoj2p

Median, Midpoint and Centroid (3)
A median intersects the opposite side at the midpoint:

https://wolfram.com/xid/0puv80fwnm-8yzr3z


https://wolfram.com/xid/0puv80fwnm-begiq1

The medians of a triangle intersect at the centroid:

https://wolfram.com/xid/0puv80fwnm-etitue


https://wolfram.com/xid/0puv80fwnm-uok388


https://wolfram.com/xid/0puv80fwnm-pemnpj

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

https://wolfram.com/xid/0puv80fwnm-khg8nw


https://wolfram.com/xid/0puv80fwnm-621e1x

Perpendicular Bisector, Midpoint and Circumcenter (3)
The perpendicular bisector of a side passes through the midpoint of that side:

https://wolfram.com/xid/0puv80fwnm-h2ko82


https://wolfram.com/xid/0puv80fwnm-jgjq9g

The perpendicular bisectors of a triangle intersect at the circumcenter:

https://wolfram.com/xid/0puv80fwnm-s32aow


https://wolfram.com/xid/0puv80fwnm-hik7xg


https://wolfram.com/xid/0puv80fwnm-7oiji6

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

https://wolfram.com/xid/0puv80fwnm-7rw40i


https://wolfram.com/xid/0puv80fwnm-0mjlkq

Altitude, Foot and Orthocenter (2)
The foot of an altitude is the intersection of the altitude and the opposite side:

https://wolfram.com/xid/0puv80fwnm-6h1in4


https://wolfram.com/xid/0puv80fwnm-lqun39

The altitudes of a triangle intersect at the orthocenter:

https://wolfram.com/xid/0puv80fwnm-cun18w


https://wolfram.com/xid/0puv80fwnm-eo83mi


https://wolfram.com/xid/0puv80fwnm-dm6vgv

Symmedian, Median and Angle Bisector (3)
The endpoint of a symmedian at a vertex is the intersection of the symmedian and the opposite side:

https://wolfram.com/xid/0puv80fwnm-3557r9


https://wolfram.com/xid/0puv80fwnm-myh7qc

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

https://wolfram.com/xid/0puv80fwnm-7mbuut


https://wolfram.com/xid/0puv80fwnm-0z5hxx


https://wolfram.com/xid/0puv80fwnm-57jtge

The symmedians of a triangle intersect at the symmedian point:

https://wolfram.com/xid/0puv80fwnm-z4ayvb


https://wolfram.com/xid/0puv80fwnm-i15lsb


https://wolfram.com/xid/0puv80fwnm-tipkot

Exterior Angle Bisector and Excenter (2)
The excenter opposite a vertex is the intersection of the exterior angle bisectors of the opposite angles:

https://wolfram.com/xid/0puv80fwnm-wk8tv7


https://wolfram.com/xid/0puv80fwnm-tigenk


https://wolfram.com/xid/0puv80fwnm-8ckx1c


https://wolfram.com/xid/0puv80fwnm-ijawbu

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

https://wolfram.com/xid/0puv80fwnm-lh74d4


https://wolfram.com/xid/0puv80fwnm-u3o219

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:

https://wolfram.com/xid/0puv80fwnm-tz2iev


https://wolfram.com/xid/0puv80fwnm-tln5l2


https://wolfram.com/xid/0puv80fwnm-jcwrj9


https://wolfram.com/xid/0puv80fwnm-pzi0q6


https://wolfram.com/xid/0puv80fwnm-55cyq9


https://wolfram.com/xid/0puv80fwnm-2nq410


https://wolfram.com/xid/0puv80fwnm-kvf0xh

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

https://wolfram.com/xid/0puv80fwnm-c0dkwv


https://wolfram.com/xid/0puv80fwnm-oep5t6

Euler Line, Centroid, Circumcenter, Orthocenter and Nine-Point Center (1)
Midpoint (1)
TriangleCenter[{a,b,c},"Midpoint"] is equivalent to Midpoint[{a,c}]:

https://wolfram.com/xid/0puv80fwnm-j55usf


https://wolfram.com/xid/0puv80fwnm-tkzn33

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
]}
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
]}