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12/9/2012
Theorem
The
perpendicular
bisectors of the
sides of a triangle
are concurrent at
a point
equidistant from
the
th vertices
ti
B
Diagram
Symbols
Perpendicular
bisectors
A
PX , PY , and PZ
are concurrent at P.
X
Y
PA = PB = PC
P
Z
C
1
12/9/2012
€
When 3 or more lines intersect at one point,
they are concurrent. The point at which they
intersect is the point of concurrency.
€
The point of concurrency of the perpendicular
bisectors of a triangle is called the circumcenter
of the triangle.
€
Since the circumcenter is equidistant from the
vertices you can use the circumcenter as the
vertices,
center of the circle that contains each vertex of
the triangle. You say the circle is circumscribed
about the triangle.
A
X
Y
P
B
Z
C
2
12/9/2012
€ Find
the circumcenter of the following 3
points.
Circumcenter
Theorem
The bisectors of
the angles of a
triangle are
concurrent at a
point equidistant
from the sides of
the triangle.
Diagram
Symbols
Angle bisectors
AP, BP, and CP
A
are concurrent at P.
PX = PY = PZ
X
Y
P
B
Z
C
3
12/9/2012
The point of concurrency of the angle bisectors
of a triangle is called the incenter of the
triangle.
€ P is the center of the circle that is inscribed in
the triangle.
A
€
X
Y
P
B
C
Z
€ If
P is the incenter and PY = 2x-4 and PZ =
5x-13, find BP, if PB = 3x.
2 x − 4 = 5 x − 13
9 = 3x
A
PB = 3(3)
X
3x
3= x
Y
P
PB = 9
2x-4
5x-13
B
Z
C
4
12/9/2012
€ P.
305 #’s 8-12 even, 15-18, 22, 33-34, 37-40
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