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Concurrent Lines, Medians,
Altitudes
Point of Concurrency
When three or more lines intersect, they are concurrent.
All triangles have 4 sets of lines
that are concurrent.
Angle Bisectors :
Point of concurrency is the incenter.
Draw
congruency
marks.
Theorem: The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides.
Angle Bisectors :
Point of concurrency is the incenter.
Draw
congruency
marks.
A circle can be
inscribed in a
triangle from the
incenter .
Theorem: The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides.
Perpendicular Bisectors:
Perpendicular Bisectors:
Point of concurrency is the circumcenter.
Draw
congruency
marks.
A circle can be
circumscribed
around a
triangle from
the
circumcenter.
Theorem: The perpendicular bisectors of the
sides of a triangle are concurrent at a point that is
equidistant from the vertices.
Median – a segment whose endpoints are a vertex and the
midpoint of the opposite side
Draw
congruency
marks.
The Centroid is the point of concurrency of medians
x
2x
Theorem– The medians of
a triangle are concurrent at
a point that is two thirds
the distance from each
vertex to the midpoint of
the opposite side.
If Q is the centroid of triangle ABC and XQ  3
XA 
QA 
CZ  6 x  9 y
QZ 
QC 
a)2 x  9 y
b )6 x  9 y
c)2 x  3 y
d )4 x  6 y
Altitude – the perpendicular segment from a vertex to the
line containing the opposite side. An altitude can be a side
of a triangle, or it can be outside of the triangle.
Theorem: The lines that contain the altitudes of a triangle are concurrent.
The Orthocenter is the point of concurrency for altitudes
The Orthocenter of an obtuse
triangle is always outside of the
triangle.
b.
a.
e.
c.
d.
Name these lines:
b.
a.
a.
b.
c.
d.
Angle bisector
Median
Altitude
Perpendicular
bisector
e. Midsegment
e.
c.
d.
AD  BC
GF  BC
BAE  CAE
BF  FC

G
Markthe
theinformation
informationgiven.
given. Identify
Identifythe
thefollowing:
following:
Mark
Median - AE
Median
Altitude - AD
Altitude
AngleBisector
Bisector-AE
Angle
PerpendicularBisector
Bisector FG
Perpendicular

G
Homework: Page276, 277: 8, 11-16 all, 19-22 all, 28 a,b,c.
Complete graph
QUIZ Monday: Distance of a line segment, midpoint of a line
segment, altitude, midsegment, angle bisector, perpendicular
bisector, medians.
Recall:
rise
slope 
run
y2  y1
m
x2  x1