Special Segments and Triangle Centers

MIDSEGMENT

A segment that joins the midpoints of two sides of a triangle.

PERPENDICULAR BISECTOR

A line, segment, or ray that divides a segment into two equal parts and is perpendicular to the segment.

ANGLE BISECTOR

A line, segment, or ray that divides an angle into two equal parts.

MEDIAN

A segment that connects a vertex of a triangle to the midpoint of the opposite side.

ALTITUDE

A segment that connects a vertex of a triangle to the opposite side so that it is perpendicular to that side.

ORTHOCENTER

The point at which the three altitudes intersect in a triangle.

CENTROID

The point at which the three medians of a triangle intersect.

CIRCUMCENTER

The point at which the three perpendicular bisectors of the sides of a triangle intersect.

INCENTER

The point at which the three angle bisectors of a triangle intersect.

ALTITUDE (height)

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

PROPERTIES OF MEDIANS

The three medians of a triangle intersect at a point called the centroid.

PROPERTIES OF ALTITUDES

The three altitudes of a triangle intersect at a point called the orthocenter.

PROPERTIES OF CIRCUMCENTER

A circumcenter is equidistant from the vertices of a triangle.

PROPERTIES OF INCENTER

An incenter is equidistant from the sides of a triangle.

MEDIAN FORMULA

If P is the centroid of triangle AJKL, JK = 22, KN = 13, and OL = 18, find each missing measure.

ALTITUDE EXAMPLES

List the altitudes: BF, DC, AF.

CENTROID EXAMPLES

If G is the centroid of triangle AACE, AG = 8, GF = 7, and BG = 5, find each missing measure.

CENTROID MEASURES

If X is the centroid of triangle ARST, TU = 27, SW = 18, and RV = 21, find each missing measure.

MEDIAN EXAMPLES

The three medians of a triangle intersect at a point called the centroid.

ALTITUDE POSITION

Altitudes can be inside a triangle, outside a triangle, or a side of the triangle.

ANGLE BISECTOR DEFINITION

The point at which the three angle bisectors intersect in a triangle.

PERPENDICULAR BISECTOR DEFINITION

The point at which the three perpendicular bisectors intersect in a triangle.