Point of concurrency
CIRCUMCENTER
INCENTER
What forms the point
of concurrency
Perpendicular
bisector: the
segment/line that
passes through the
midpoint and forms a
right angle
Angle bisector: the
segment/line that
divides the angle in
half
CENTROID
(center of gravity)
Median: segment
connecting the vertex
to the opposite side’s
midpoint
ORTHOCENTER
Altitude (Height): the
segment that passes
through a vertex and is
perpendicular to the
opposite side
Location of the point
of concurrency
Acute – inside the
triangle
Obtuse – outside the
triangle
Right – midpoint of
the hypotenuse
Acute – inside the
triangle
Obtuse – inside the
triangle
Right – inside the
triangle
Acute – inside the
triangle
Obtuse – inside the
triangle
Right – inside the
triangle
Acute – inside the
triangle
Obtuse – outside the
triangle
Right – at the right
angle
Traits of the point of
concurrency
Circumcenter is the
center of the
circumscribed circle.
The radii goes from
the circumcenter to
the vertex.
Incenter is the center
of the inscribed circle.
The radii goes from
the incenter to the side
of the triangle.
The centroid is 2/3 the
median from the
centroid to the vertex;
and 1/3 the median
from the centroid to
the side of the triangle.
-----------------------------
How to find the circumcenter algebraically:
1) find the midpoint of the side of the triangle
2) find the slope of a triangle and use its opposite reciprocal
3) write the equation in point-slope form using the midpoint and the opposite
reciprocal
4) repeat for the other three sides
5) use two of the equations to find the point of intersection using either
substitution or elimination
6) check the point in the equation not used
How to find the orthocenter algebraically:
1) determine which vertex you are working from and the its coordinates
2) find the slope of the side opposite the chosen vertex and use its opposite
reciprocal
3) write the equation in point-slope form using the midpoint and the opposite
reciprocal
4) repeat for the other three sides
5) use two of the equations to find the point of intersection using either
substitution or elimination
6) check the point in the equation not used
How to find the centroid: average the three points