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Section 1.5
Special Points in Triangles
CONCURRENT
• The point where 3 or more lines intersect
TRIANGLES & CIRCLES
What are some things you already know about triangles or circles?
• How many sides does a triangle have?
• What kinds of triangles are there?
PREVIOUS FACTS…
WHAT IS A PERPENDICULAR
BISECTOR??
PERPENDICULAR BISECTOR
• Forms right angles.
• Splits the segment in half
PERPENDICULAR BISECTORS & TRIANGLES
Now let’s combine perpendicular bisectors and
triangles.
BEGIN BY…• Finding the
perpendicular bisector of each side of the triangle.
WHAT CAN WE CONCLUDE?
• What happens when these 3 lines intersect? What is formed?
Perpendicular Bisectors
• The perpendicular bisectors of a triangle ____________ at a single __________.
EXAMPLE
CIRCUMCENTER
• The intersection point of the perpendicular bisectors of a triangle
Circumcenter of an Obtuse Triangle
WHAT IS A ANGLE
BISECTOR??
• Splits the angle in half
ANGLE BISECTOR
ANGLE BISECTORS & TRIANGLES
Now let’s combine angle bisectors and triangles.
BEGIN BY…• Finding the
angle bisector of each angle of the triangle.
WHAT CAN WE CONCLUDE?
• What happens when these 3 lines intersect? What is formed?
Angle Bisector
• The angle bisectors of a triangle __________ at a single _________.
EXAMPLE
INCENTER
• The intersection point of the angle bisectors of a triangle
INSCRIBED & CIRCUMSCRIBED
CIRCLES
INSCRIBED
• Inside the triangle
• Just touches the three sides of the triangle.
• Another definition of Incenter:
• The center of an inscribed circle
INCENTER
CIRCUMSCRIBED
• Outside the triangle
• Contains all 3 vertices
• The center of a circumscribed circle
CIRCUMCENTER
A couple of other
definitions you need to know
Altitude
• A perpendicular line segment from a vertex of a triangle to the line containing the opposite side.
• How many altitudes can a triangle have?
Median
• A line segment from a vertex to the midpoint of the opposite side.
Centroid
• The point where the 3 medians meet