Bisectors of Triangles, Medians, and Altitudes

Perpendicular Bisector of a Triangle

a line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side

Circumcenter

the point of concurrency of the perpendicular bisectors of a triangle

Circumscribed Circle of a Triangle

The circle created by the vertices of the triangle, with the circumcenter as the center of the circle.

Concurrency of Perpendicular Bisectors Theorem

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

Circumcenter of Obtuse Triangle

outside the triangle

Circumcenter of Acute Triangle

inside the triangle

Circumcenter of Right Triangle

on the hypotenuse of the triangle

Angle Bisector of a Triangle

a bisector of an angle of the triangle

Incenter of a Triangle

the point of concurrency of the angle bisectors of a triangle

Inscribed Circle of a Triangle

A circle inside the triangle with the incenter as the center of the circle.

Concurrency of Angle Bisectors of a Triangle Theorem

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

Median of a Triangle

a segment from a vertex to the midpoint of the opposite side

Centroid of a Triangle

The point of concurrency of the medians of a triangle

Concurrency of Medians 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.

Altitude of a Triangle

a perpendicular segment from a vertex to the line containing the opposite side

Orthocenter of a Triangle

the point of concurrency of the three altitudes of a triangle

Orthocenter of Obtuse Triangle

outside the triangle

Orthocenter of Acute Triangle

inside the triangle

Orthocenter of Right Triangle

On the vertex of the right angle

What points of concurrency are always inside the triangle?

Centroid and Incenter

What points of concurrency can be outside or on the triangle?

Orthocenter and Circumcenter