area of the tr
iangle in t
w
o.
1
2
Midpoint
of AB
meet the opposi
te side, and
are per
p
endicul
ar t
o tha
t side.
Alti
tude on BC
A line f
rom the incenter
t
o a v
ertex exac
tl
y bisec
ts
the angle a
t tha
t v
ertex.
triangle centers
A triangle has more than one center
Believe it or not, there are four common ways of finding the “center” of a triangle.
A
All thr
ee
inter
sec
t a
t the
Midpoint of AC
centroid.
CENTROID
The centroid is the intersection of all
three medians (lines from the midpoint
of a side to the opposite vertex).
T
hese lines ar
e ca
lled
MEDIANS—they chop the
C B
Midpoint of BC
A
ORTHOCENTER
The orthocenter is the intersection
of the altitudes drawn on all three
sides.
C
Alti
tude
on AB
Alti
tude on AC
Alti
tudes go f
rom a v
ertex t
o
B
INCENTER
3
The incenter is the center of the
biggest circle you can draw inside
the triangle.
Draw a circle inside which
just touches all three sides.
“incircle”
Incenter is center of the “incircle.”
182 Chapter 4
C
ir
cl
e
d
r
a
w
n o
ut
s
id
e
through a
ll three
A line f
rom the circumc
enter
t
o the midp
oint of an
y side
meets tha
t side a
t 90º.
triangle properties
4
CIRCUMCENTER
The circumcenter is the center of the
smallest circle you can draw around
the outside of a triangle.
Circumc
enter is the
center of this circle.
vertic
es.
Ca
lled “circumscr
ibing”
Centroid
“The point nearest to all
three fences”
Which kind of center was each of your buddies describing?
Pair ’em up! (You’ll have one center left over.)
Orthocenter
“The point nearest to all
three corners”
“The point where half the
crowd is on either side, no
matter which way you look”
Circumcenter
Incenter
you are here 4 183
match up the centers
Centroid
“The point nearest to all
three fences”
Which kind of center was each of your buddies describing?
Pair ’em up! (You’ll have a center left over.)
SOlUTion
Orthocenter
“The point nearest to all
three corners”
“The point where half the
crowd is on either side, no
matter which way you look”
Circumcenter
Incenter
There must be a reason.
Geometry people never do anything without
a reason. So, if there are different ways of
defining the center it suggests that there must
be times when one center is more appropriate
than another, don’t you think?
Really, four different
centers? Some of those
don’t really look like a
“center” at all. What gives?
184 Chapter 4
triangle properties
Geometry Detective
Below are four different triangles. Work out (roughly) where each of
the four types of “center” is for each.
Are there any definitions of “center” that wouldn’t always work for
placing the drinks stall?
you are here 4 185
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