incenter of the triangle

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First found May 22, 2018

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```Math 1
Warm-ups
1-8-09
Fire stations are located at A and B. XY , which contains
Havens Road, represents the perpendicular bisector of AB
.
A fire is reported at point X. Which fire station is
closer to the fire? Explain.
Math 1
1-9-09
Warm-ups
What is the incenter of a triangle?
A median of a triangle is a segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side.
Every triangle has three medians, and the medians
are concurrent.
An altitude of a triangle is a perpendicular segment
from a vertex to the line containing the opposite side.
Every triangle has three altitudes. An altitude can be
inside, outside, or on the triangle.
Math Support
1-8-09
Warm-ups
True or false:
An altitude of a triangle is a perpendicular
segment from a vertex to the line containing the
opposite side.
Vocabulary
concurrent
point of concurrency
circumcenter of a triangle
incenter of a triangle
Orthocenter
Centroid
Point of concurrency
When three or more lines intersect at one point, the
lines are said to be concurrent. The point of
concurrency is the point where they intersect.
Incenter of a circle
A triangle has three angles, so it has three angle
bisectors. The angle bisectors of a triangle are also
concurrent. This point of concurrency is the incenter
of the triangle .
The incenter is always inside the triangle.
Incenter of a circle
A triangle has three angles, so it has three angle
bisectors. The angle bisectors of a triangle are also
concurrent. This point of concurrency is the incenter
of the triangle .
The incenter is always inside the triangle.
Incenter of a circle
The incenter is the center of the triangle’s inscribed circle.
A circle inscribed in a polygon intersects each line that
contains a side of the polygon at exactly one point.
Point of concurrency is
center of imaginary
inscribed circle
Incenter is equidistant from the sides
PX = PY = PZ
Incenter Example:
Using Properties of Angle Bisectors
MP and LP are angle bisectors of ∆LMN. Find the
distance from P to MN.
P is the incenter of ∆LMN. By the Incenter Theorem,
P is equidistant from the sides of ∆LMN.
The distance from P to LM is 5. So the distance
from P to MN is also 5.
Circumcenter center of a circle
The three perpendicular bisectors of a triangle are
concurrent. This point of concurrency is the
circumcenter of the triangle.
The circumcenter can be inside the triangle,
outside the triangle, or on the triangle.
Circumcenter of a triangle
The three perpendicular bisectors of a triangle are
concurrent. This point of concurrency is the circumcenter
of the triangle.
The circumcenter can be inside, outside, or on the triangle.
Point of concurrency is center of
circumscribed circle
Circumcenter is equidistant
from the vertices
PA = PB = PC
Circumcenter Example:
Using Properties of Perpendicular Bisectors
DG, EG, and FG are the
perpendicular bisectors of
∆ABC. Find GC.
G is the circumcenter of ∆ABC. By
the Circumcenter Theorem, G is
equidistant from the vertices of
∆ABC.
GC = GB
GC = 13.4
Circumcenter Thm.
Substitute 13.4 for GB.
Centroid of a Triangle
The point of concurrency of the median of a triangle
is the centroid of the triangle . The centroid is
always inside the triangle. The centroid is also
called the center of gravity because it is the point
where a triangular region will balance.
AP to PY is 2:1
CP to PX is 2:1
ZP to PB is 2:1
Centroid Example
Using the Centroid to Find Segment Lengths
In ∆LMN, RL = 21 and SQ =4.
Find LS.
Centroid Thm.
Substitute 21 for RL.
LS = 14
Simplify.
The Orthocenter of a Circle
The point of concurrency of the altitudes of a
triangle is the orthocenter of the triangle.
The Orthocenter of a Circle
In ΔQRS, altitude QY is inside the triangle, but RX
and SZ are not. Notice that the lines containing the
altitudes are concurrent at P. This point of
concurrency is the orthocenter of the triangle.
```