Geometry
6.2 & 6.3 Triangle Concurrency Points
Essential Question
What are points of concurrency?
January 14, 2016
6.2 and 6.3 Points of Concurrency
Goals
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Review terms.
Know what medians and altitudes are.
Review characteristics of each concurrent
point.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Lines
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Three or more lines intersecting at the same
point are concurrent.
The point where they intersect is the point of
concurrency.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Perpendicular Bisector
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The perpendicular bisector of a line segment
is the line that is perpendicular to the
R
segment at its midpoint.
A
B
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Angle Bisector
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The angle bisector is a ray that divides an
angle into two congruent adjacent angles.
A
D
C
B
January 14, 2016
6.2 and 6.3 Points of Concurrency
Altitude
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An altitude of a triangle is a segment drawn
from a vertex perpendicular to the opposite
side (or to the line containing the opposite
side).
A triangle has three altitudes.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Median
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A median of a
triangle is the
segment
drawn from a
vertex to the
midpoint of the
opposite side.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Points of Concurrency
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Four points of concurrency:
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Circumcenter
Incenter
Centroid
Orthocenter
Made using different types of lines and have
a variety of properties.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Circumcenter
Created when using the perpendicular
bisectors of each side of a triangle.
In the example box, draw one of the
perpendicular bisectors of the triangle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Circumcenter
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When all three perpendicular bisectors are
drawn, the point of concurrency created is
called the circumcenter.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Circumcenter Property
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The circumcenter is equidistant from each
vertex of the triangle.
This is called a
circumcircle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Incenter
Created when using the angle bisectors of each
vertex of a triangle.
In the example box, draw one of the angle
bisectors of the triangle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Incenter

When all three angle bisectors are drawn, the
point of concurrency created is called the
incenter.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Incenter Property
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The incenter is equidistant from the sides of a
triangle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Incenter Property
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The incenter is equidistant from the sides of a
triangle.
This is called
an incircle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Centroid
Created when using the medians of a triangle.
In the example box, draw one of the medians of
the triangle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Centroid
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When all three medians are drawn, the point
of concurrency created is called the centroid.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Centroid Property
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The centroid of a triangle is two thirds of the
distance from each vertex to the midpoint of
the opposite side.
A
AB  AC
2
3
BC  AC
1
3
B
C
January 14, 2016
6.2 and 6.3 Points of Concurrency
Centroid Property
2
x
3
x
1
x
3
January 14, 2016
6.2 and 6.3 Points of Concurrency
Examples
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Turn to your notes to practice a few
examples.
C is the centroid of the triangle.
R
C
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 1
RS = 9
R
RC = ?
CS = ?
6
C
3
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 2
RS = 15
R
RC = ?
CS = ?
10
C
5
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 3
RS = ?
R
RC = 8
CS = ?
8
12
C
4
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 4
RS = ?
R
RC = 30
CS = ?
45
30
C
15
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 5
RS = ?
R
RC = ?
CS = 7
21
14
C
7
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Example 6
RS = ?
R
RC = ?
CS = 4.2
12.6
8.4
C
4.2
S
January 14, 2016
6.2 and 6.3 Points of Concurrency
Centroid Property

The centroid of a
triangle is also known
as the center of
balance.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrent Point: Orthocenter
Turn back to the graphic organizer.
Created when using the altitudes of a triangle.
In the example box, draw one of the altitudes of
the triangle.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Concurrency Point: Orthocenter

When all three altitudes are drawn, the point of
concurrency created is called the orthocenter.
January 14, 2016
6.2 and 6.3 Points of Concurrency
Orthocenter Property
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None!
January 14, 2016
6.2 and 6.3 Points of Concurrency
Fast answers!
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The altitudes are concurrent at the ?
Orthocenter
The medians are concurrent at the ?
Centroid
The perpendicular bisectors are concurrent at
the ?
Circumcenter
The angle bisectors are concurrent at the ?
Incenter
January 14, 2016
6.2 and 6.3 Points of Concurrency
Fast answers!
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Which point is equidistant from the sides of a
triangle?
Incenter
Which point is the center of balance?
Centroid
Which point is equidistant from the vertices?
Circumcenter?
January 14, 2016
6.2 and 6.3 Points of Concurrency
Fast answers!
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What point is needed to draw a circumcircle?
Circumcenter
What point is needed to draw an incircle?
Incenter
What point is needed to find the center of
balance?
Centroid
January 14, 2016
6.2 and 6.3 Points of Concurrency
Do you know…
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What a median is?
How to draw a perpendicular bisector?
What lines are needed to find the incenter?
How to locate the circumcenter?
What point is located using the medians?
How to construct an altitude?
Which concurrent point is the same distance
from each vertex of a triangle?
January 14, 2016
6.2 and 6.3 Points of Concurrency