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[TriangleCenter[{a,b,c},"Incenter"],TriangleMeasurement[{a,b,c},"Inradius"]]:

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[TriangleCenter[{a,b,c},"Circumcenter"],TriangleMeasurement[{a,b,c},"Circumradius"]]:

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"]:

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)

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:

Introduced in 2019
 (12.0)