5-3 Points of Concurrency

1
5-3Points of Concurrency

• Objective
• To identify properties of perpendicular bisectors
  and angle bisectors

2
Concurrent When three or more lines, segments,
rays or planes have a point in common. Point of
Concurrency The point of intersection.
Point of Concurrency
Concurrent
Not Concurrent
3
The three perpendicular bisectors of a triangle
are concurrent. Point of concurrency is called
circumcenter.
The circumcenter of a triangle is equidistant
from the vertices and is the center of the
circle.
4
The circumcenter of a triangle can be inside, on
, or outside a triangle.
5
Finding the Circumcenter of a triangle.
6
The point of concurrency of the angle bisectors
of a triangle is called the incenter of the
triangle.
p. 305 7, 8, 9, 15, 17, 37, 38
The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides
of the triangle.
P is the center of the circle that is inscribed
in the triangle.
7
The circle is circumscribed about the triangle.
Circumcenter
8
Angle Bisector Concurrency Conjecture
The three angle bisectors of a triangle are
concurrent. Point of concurrency is called
incenter.
Incenter Conjecture
The incenter of a triangle is equidistant from
the sides.
Inscribed Circle
Incenter
9
Altitude Concurrency Conjecture
The three altitudes (or the lines containing the
altitudes) of a triangle are concurrent. Point
of concurrency is called orthocenter.
Pg 179 1-4, 6
Orthocenter
10
Group Construct
1 Each angle bisectors for an acute triangle
2 Each angle bisectors for an obtuse triangle
3 Each perpendicular bisectors for an acute triangle
4 Each perpendicular bisectors for an obtuse triangle
5 Each altitude for an acute triangle
6 Each altitude for an obtuse triangle

• Directions
• Each group member does assigned construction.
• Compare and discuss all constructions
• Write a conjecture about your construction
• Measure from point of concurrency to vertices and
  sides
• Discuss findings and write a conjecture.
• Elect spokes person to share construction and
  conjectures