# TriangleConstruct

TriangleConstruct[tri,type]

gives the specified type of construct for the triangle tri.

# Details    • TriangleConstruct can give a Point, Line, InfiniteLine, Circle or Triangle object.
• The triangle tri can be given as {p1,p2,p3}, Triangle[{p1,p2,p3}] or Polygon[{p1,p2,p3}].
• • The following point 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
• • The following line types can be given:
•  {"Altitude",p} altitude passing through the vertex p {"AngleBisectingCevian",p} cevian bisecting the interior angle at the vertex p {"AngleBisector",p} bisector of the interior angle at the vertex p "Boundary" boundary {"Cevian",center,p} cevian passing through the vertex p and the specified center "EulerLine" Euler line {"ExteriorAngleBisector",p} bisector of the exterior angle at the vertex p {"Median",p} median passing through the vertex p {"OppositeSide",p} side opposite from the vertex p {"PerpendicularBisector",p} perpendicular bisector of the side opposite from p {"Symmedian",p} symmedian passing through the vertex p
• • The following circle types can be given:
•  "Circumcircle" circumscribed circle {"Excircle",p} excircle opposite from the vertex p "Incircle" inscribed circle "NinePointCircle" nine-point circle
• • The following triangle types can be given:
•  "AntimedialTriangle" antimedial triangle "MedialTriangle" medial triangle "Triangle" original triangle
• 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 forms {"CevianEndpoint",center,p} and {"Cevian",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.
• TriangleConstruct can be used with symbolic points in GeometricScene.

# Examples

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## Basic Examples(2)

Calculate the altitude of a triangle:

Calculate the excenter of a triangle at the specified vertex:

## Scope(29)

### Points(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:

### Lines(10)

Calculate the altitude of a triangle:

Calculate the angle bisector of a triangle:

Get the angle bisector as a line segment:

Calculate the boundary of a triangle:

Calculate a cevian passing through the orthocenter:

Calculate the cevian passing through a different vertex:

Calculate a cevian through an arbitrary center point:

Calculate the Euler line of a triangle:

Calculate the exterior angle bisector at a vertex:

Calculate a median of a triangle:

Calculate the side opposite from a specified vertex:

Calculate the perpendicular bisector of a side of a triangle:

Calculate a symmedian:

### Circles(4)

Calculate the circumcircle of a triangle:

Calculate the excenter of a triangle at the specified vertex:

Calculate the incircle of a triangle:

Calculate the nine-point center of a triangle:

### Triangles(3)

Calculate the antimedial triangle of a triangle:

Calculate the medial triangle of a triangle:

Obtain the original triangle:

## Properties & Relations(28)

### Angle Bisector and Incenter(5)

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

TriangleConstruct[{a,b,c},"Incircle"] is equivalent to Circle@@Insphere[{a,b,c}]:

### Median, Midpoint and Centroid(3)

A median intersects the opposite side at the midpoint:

The medians of a triangle intersect at the centroid:

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

### Perpendicular Bisector, Midpoint and Circumcenter(5)

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},"PerpendicularBisector"] is equivalent to PerpendicularBisector[{a,c}]:

TriangleConstruct[{a,b,c},"Circumcircle"] is equivalent to Circle@@Circumsphere[{a,b,c}]:

### 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(3)

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

TriangleConstruct[{a,b,c},"ExteriorAngleBisector"] is equivalent to AngleBisector[{a,b,c},"Exterior"]:

### 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:

### 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)

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

### Boundary(1)

TriangleConstruct[{a,b,c},"Boundary"] is equivalent to RegionBoundary[Triangle[{a,b,c}]]:

### Medial and Antimedial Triangle(2)

TriangleConstruct[{a,b,c},"MedialTriangle"] is equivalent to Triangle[TriangleCenter[tri,{"Midpoint",All}]]:

The antimedial triangle is the triangle whose medial triangle is the original triangle: