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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