Relationships in Triangles

Median

A segment joining a vertex to the midpoint of the opposite side

Altitude

A segment joining a vertex to the opposite side so that it is perpendicular to that side

Converse of Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Angle Bisector Theorem

If a point is on a bisector of an angle, then the point is equidistant from the sides of the angle.

Perpendicular Bisector Theorem

If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse of Angle Bisector Theorem

If a point, is on the interior of an angle and equidistant from the sides of the angle, then the point is on the angle bisector.

Angle Bisector

In a triangle, a line, a segment, or a ray that divides an angle into two congruent angles.

Perpendicular Bisector

In a triangle, a line, a segment, or a ray that passes through the midpoint of a side and is perpendicular to that side

Orthocenter

The altitudes of the angles of a triangle intersect at a point

Incenter

The angle bisectors of the angles of a triangle intersect at a point

Centroid Theorem

The centroid is 2/3 of the distance from each vertex to the midpoint of the opposite side.

Circumcenter Theorem

The circumcenter is always equidistant from the vertices of the triangle.

Incenter Theorem

The incenter is always equidistant from the sides of the triangle.

Centroid

The medians of the angles of a triangle intersect at a point

Circumcenter

The perpendicular bisectors of the sides of a triangle intersect at a point