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Geometric Constructions: Points and Lines of Concurrency in Triangles, Study notes of Analytical Geometry

DeVry University - IndianaAnalytical Geometry

An overview of various geometric constructions related to triangles, including the incenter, circumcenter, centroid, and orthocenter. It explains how to find these points and describes the properties of their respective bisectors and altitudes. Additionally, the document introduces the concept of concurrent lines and the point of concurrency.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

seshadrinathan_hin
seshadrinathan_hin 🇺🇸

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Page 1 of 2
Points of Concurrency
Point of Concurrency
Formed by
Used to
Contains
Triangle
Vertex
Pt. inside/outside
of triangle
Sketch
Incenter
Angle
Bisectors of
Triangle
Inscribe a circle
in a triangle
Find a point
equidistant from all
3 sides
Yes
Always inside
Circumcenter
Perpendicular
Bisectors of
Triangles
Circumscribe a
circle about a
triangle
Find point
equidistant from all
3 vertices
Sometimes
Acute - Inside
Right - On
Obtuse -
Outside
Centroid
Medians of
Triangle
Find the
center of gravity
of the triangle
Yes
Always inside
Orthocenter
Altitudes of
Triangle
??
Yes
Acute - Inside
Right - On
Obtuse -
Outside
pf2

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Page 1 of 2

Points of Concurrency

Point of Concurrency Formed by Used to Contains Triangle Vertex Pt. inside/outside of triangle Sketch

Incenter

Angle

Bisectors of

Triangle

 Inscribe a circle in a triangle  Find a point equidistant from all 3 sides

Yes Always inside

Circumcenter

Perpendicular

Bisectors of

Triangles

Circumscribe a circle about a triangle Find point equidistant from all 3 vertices

Sometimes

 Acute  - Inside  Right  - On  Obtuse  - Outside

Centroid

Medians of

Triangle

Find the

center of gravity

of the triangle

Yes Always inside

Orthocenter

Altitudes of

Triangle

?? Yes

 Acute  - Inside  Right  - On  Obtuse  - Outside

Page 2 of 2 Angle Bisector of a triangle – A segment from the vertex of the triangle that bisects that angle. Perpendicular Bisector of a triangle – A segment that is perpendicular to a side of that triangle at its midpoint. Median of a triangle – A segment that connects the vertex of the triangle to the midpoint of the opposite side. Altitude of a triangle – A segment from the vertex of a triangle that is perpendicular to the opposite side. Concurrent Lines – Three of more coplanar lines that intersect at the same point. Point of concurrency – The point where the concurrent lines intersect.