Triangle Centers in Geometry
Section 7.7 Circumcenter,
Incenter, Orthocentre, and
Centroid of a triangle
I) Intersection of Angle Bisectors [Incenter]
•
An angle bisector is a line that splits an
angle in half
•
What happens when you draw all three
angle bisectors of a triangle and connect
them?
•
The intersection of all three angle bisectors
will be the center of the inscribed circle
•
This point is known as the “Incenter”
•
The radius of this circle is perpendicular to
each side of the triangle
Incenter
II) Perpendicular Bisectors of Each Side
•
A “
perpendicular bisector”
is a line that
cuts a line in half and is perpendicular to it
•
When happens when your draw the
“
perpendicular bisectors”
of each side
•
The intersection of all three perpendicular
bisectors will be the center of a circle that
circumscribes this triangle
•
This point is called the “Circumcenter”
•
The radius of this circumscribed circle will
be the center to the vertices of the triangle
circumcenter
III) Medians of all three sides
•
A median is a line that connects a vertex to
the midpoint of the opposite side
•
What happens when you draw the median of
all three sides?
•
The intersection of all three medians is called
the “centroid”
•
The centroid is the center of gravity of this
circle
•
The distance of the centroid of any side
is half the distance from the opposite vertex
Centroid
IV) Altitudes of all three sides
•
The altitude is perpendicular to one side and
connects to the opposite vertex
•
What happens when you connect all three altitudes?
•
The intersection of all three altitudes is a point called
the “orthocenter”
•
The orthocenter is useless
•
If the triangle is obtuse, then the orthocenter is
outside of the circle
Orthocenter
Challenge: In the diagram, CD is a diameter. Angle ECD is 50 degrees, angle
EAD is also 25 degrees. What is angle DAB? Note(CEA and CFB are straight
lines)
(CNML)
∆ AMN & ∆ ABC are similar triangles
Use trig. to get the area of ∆ABC
Draw the inscribed circle for ideas
Get the altitude from point A
and the radius of the circle
AMC 12A 2011 #13
Aime I) 2016
In triangle ABC, let “I” be the center of
the inscribed circle, and let the
bisector of angle ACB intersect AB at
“L”. The line through “C” and “L”
intersect the circumscribed circle of
triangle ABC at two points “C” and “D”.
If LI=2, and LD = 3, then IC=p/q, where
“p” and “q” are relatively prime
positive integers. Find the value of
p+q:
Find all the similar triangles!
Amc 12a 2011
Triangle Centers, Circumcenter, Incenter, Orthocenter, Centroid, Geometry
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Presentation Transcript
Section 7.7 Circumcenter, Incenter, Orthocentre, and Centroid of a triangle
I) Intersection of Angle Bisectors [Incenter] An angle bisector is a line that splits an angle in half What happens when you draw all three angle bisectors of a triangle and connect them? The intersection of all three angle bisectors will be the center of the inscribed circle This point is known as the Incenter The radius of this circle is perpendicular to each side of the triangle A R s = p s = xx z R z y Incenter y 2
II) Perpendicular Bisectors of Each Side A perpendicular bisector is a line that cuts a line in half and is perpendicular to it When happens when your draw the perpendicular bisectors of each side The intersection of all three perpendicular bisectors will be the center of a circle that circumscribes this triangle This point is called the Circumcenter The radius of this circumscribed circle will be the center to the vertices of the triangle circumcenter
III) Medians of all three sides A median is a line that connects a vertex to the midpoint of the opposite side What happens when you draw the median of all three sides? The intersection of all three medians is called the centroid The centroid is the center of gravity of this circle The distance of the centroid of any side is half the distance from the opposite vertex 2x z y 2y 2z x Centroid
IV) Altitudes of all three sides The altitude is perpendicular to one side and connects to the opposite vertex What happens when you connect all three altitudes? The intersection of all three altitudes is a point called the orthocenter The orthocenter is useless If the triangle is obtuse, then the orthocenter is outside of the circle Orthocenter
Challenge: In the diagram, CD is a diameter. Angle ECD is 50 degrees, angle EAD is also 25 degrees. What is angle DAB? Note(CEA and CFB are straight lines) (CNML)
AMC 12A 2011 #13 AMN & ABC are similar triangles Use trig. to get the area of ABC Draw the inscribed circle for ideas A Get the altitude from point A and the radius of the circle N M C B
C In triangle ABC, let I be the center of the inscribed circle, and let the bisector of angle ACB intersect AB at L . The line through C and L intersect the circumscribed circle of triangle ABC at two points C and D . If LI=2, and LD = 3, then IC=p/q, where p and q are relatively prime positive integers. Find the value of p+q: Find all the similar triangles! x Aime I) 2016 2 I 2 A B L D 3
