Triangle Points of Concurrency

point of concurrency

where three lines that are concurrent intersect

centroid

point of concurrency for the medians of a triangle

circumcenter

point of concurrency for the perpendicular bisectors of a triangle

incenter

point of concurrency for the angle bisectors of a triangle

orthocenter

point of concurrency for the altitudes of a triangle

centroid, incenter

point(s) of concurrency that are always on the interior of the triangle

circumcenter, orthocenter

point(s) of concurrency that are always inside an acute triangle, outside an obtuse triangle, and on a vertex or side of a right triangle

1:2

ratio of the lengths of the segments formed by the centroid on its special segment

at the point that is located at 2/3 the length of the median

location of the centroid on the median

concurrent

happening or occurring at the same point or time

they intersect at one point.

3 lines are concurrent if...

vertex of the right angle.

In a right triangle, the orthocenter is...

midpoint of the hypotenuse.

In a right triangle, the circumcenter is...

equidistant from the vertices of the triangle

(regarding distance to certain parts of a triangle) The circumcenter is ... (from) ...

center of a circumscribed circle.

The circumcenter is called the circumcenter because it is...

equidistant from the sides of the triangle

(regarding distance to certain parts of a triangle) The incenter is ... (from) ...

the center of an inscribed circle.

The incenter is called the incenter because it is...

circumcenter, centroid, and orthocenter

point(s) of concurrency that always lie on the Euler Line

centroid,between the circumcenter and orthocenter.

On the Euler Line, this point of concurrency is always between these two points of concurrency.

1:2 (The distance from the circumcenter to the centroid is half the distance from the centroid to the orthocenter.)

ratio of the lengths of the segments on the Euler Line

incenter

Only in an isosceles triangle, the ... lies on the Euler Line

are the same point

In an equilateral triangle, the points of concurrency all ...

median

a line from the vertex of a triangle to the midpoint of the opposite side

altitude

a line from the vertex that is perpendicular to the opposing side

perpendicular bisector

a line that goes through a midpoint of a line and creates a right angle (does not have to go through the vertex)

angle bisector

a line from the vertex of an angle that bisects the angle