Bisectors, Medians and Altitudes Points of Concurrency Bisectors, Medians and Altitudes

Perpendicular Bisectors Any segment, line, or plane that intersects a segment at its midpoint and is perpendicular to the segment.

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

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

Concurrency When three or more lines intersect at a common point, the lines are called concurrent lines. The point where concurrent lines intersect is called the point of concurrency.

Circumcenter Theorem The point of concurrency for the perpendicular bisectors of a triangle is called the circumcenter and it is equidistant from the vertices of the triangle.

Circumcenter Theorem The circumcenter can be on the interior, exterior, or side of a triangle

Angle Bisectors An angle bisector divides an angle into two congruent angles. It can be a line, segment, or ray.

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

Converse of the Angle Bisector Theorem If a point is in the interior of an angle equidistant from the sides of the angle, then it is on the bisector of the angle.

Incenter Theorem The point of concurrency for the angle bisectors of a triangle is called the incenter and it is equidistant from the sides of the triangle.

Median A median of a triangle is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side.

Centroid Theorem The point of concurrency for the medians of a triangle is called the centroid and it is two thirds of the distance from each vertex to the midpoint of the opposite side.

Altitudes An altitude of a triangle is a segment from a vertex to the line containing the opposite side and is perpendicular to the line containing that side. An altitude can be in the interior, exterior, or on the side of a triangle.

Orthocenter The lines containing the altitudes of a triangle are concurrent and intersect at a point called the orthocenter.

Points of Concurrency