Lesson 5-2
Objective – To apply properties of perpendicular
bisectors and angle bisectors of a triangle.
Point of concurrency - A single point of intersection
for 3 or more lines.
Circumcenter of a triangle - The single point of
concurrency for the 3 perpendicular bisectors of a
triangle.
g
Circumcenter Theorem
The circumcenter of a triangle is equidistant from
the vertices of the triangle.
E
DC  EC  FC
C
Circumcenters can lie inside or outside of the triangle.
E
Circle C is circumsribed about DEF.
F
DEF is inscribed in Circle C.
C
D
D
F
Y
Proof of Circumcenter Theorem
Given: Lines l , m, & n are
 bisectors
Prove: CZ  CX  CY
m
l
C
C is the circumcenter of XYZ
C lies on the  bisector of XZ
(line n).
n
 C is equidistant from X and Z, by the Perpendicular
Bisector Theorem.  CZ  CX
C lies on the  bisector of XY (line l ).
 C is equidistant from X and Y, by the Perpendicular
Bisector Theorem.  CX  CY
 by the Transitive Prop of Equality, CZ  CX  CY.
X
Z
Finding the Circumcenter of a Right Triangle
C (8,6)
(4,3)
A (0
(0,0)
0)
x4
y3
B (8,0)
(8 0)
Find the distances below if EP  5.5, DR  6, QF  7,
and EC  7.3.
E
1) DF  12
5.5
P 7.3
2) CF  7.3
3) EQ  7
D
F
R
5) DC  7.3
Find the circumcenter of the triangle with the given
vertices.
1) A(-10,0), B(0,-8), C(0,0)
2) Find the coordinate
of the intersection.
A(-10,0)
The  bisectors of the legs always coincide with
the midpoint of the hypotenuse.
6
7
4) DE  11
Steps
1) Find the  bisector
of each leg.
Why is this the circumcenter?
Q
C
2) L(9,0), M(0,0), N(0,-4)
x  5
x  4.5
C
M
y  4
B(0,-8)
circumcenter  ( 5, 4)
L(9,0)
y  2
N(0,-4)
circumcenter  (4.5, 2)
The hypotenuse of a right triangle inscribed within a
circle will always coincide with the diameter of the circle.
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
1
Lesson 5-2
Incenter Theorem
The incenter of a triangle is equidistant from the
sides of the triangle.
Incenter of a triangle - The single point of concurrency
for the 3 angle bisectors of a triangle.
Y
Y
m
l
X
B
A
l
I
m
AI  BI  CI
I
Z
n
X
Z
C
n
The incenter always lies inside the triangle.
ABC is circumscribed about Circle I.
Circle I is inscribed in ABC.
Find the distances and angles below.
Y
1) CZI  24
2) CIZ  66
68
3) XYZ  68
4) IYZ  34
32
X
m
B
A
l
I
24
5.2
Z
C
n
5) Distance from Point I to YZ  5.2
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
2