Big Picture

Within a given triangle, there are many theorems involving bisectors, medians, and altitudes. Recall that a bisector is a line segment or line that divides a geometric shape into two congruent shapes. A median is a line segment that divides a triangles by joining a vertex to the midpoint of the opposite side. An altitude is a line segment that joins the vertex of a triangle perpendicularly to the opposite side.

Key Terms

Slope of a Line

For every triangle, there are three midsegments.

Furthermore, $\overline{DF}||\overline{AC},\overline{DE}||\overline{BC},\overline{FE}||\overline{BA}$

Midsegment Theorem: The midsegment of a triangle is half the length of the side it is parallel to.

• DF is half of AC, DE is half of BC, FE is half of BA

• DF is half of AC, DE is half of BC, FE is half of BA

Perpendicular Bisectors

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

• If $\overleftrightarrow{CD}$ is the  $\perp$ bisector of AB, then AC = CB

• If XXXXXX is the XXXXXX  bisector of AB, then AC = CB

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

Isosceles Triangles

Isosceles Perpendicular Bisector Theorem: The angle bisector of the vertex angle in an isosceles triangle is the perpendicular bisector to the base.

• The converse is also true: The perpendicular bisector of the base of an isosceles triangle is the angle bisector of the vertex angle.

• The converse is also true: The perpendicular bisector of the base of an isosceles triangle is the angle bisector of the vertex angle.

Perpendicular Bisectors (cont.)

Concurrency of Perpendicular Bisectors Theorem: The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the vertices.

Angle Bisectors

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

• If $\overrightarrow{BD}$ bisects  ∠ABC, BE ⊥  ED, and BF ⊥  FD, then ED = FD.
• The shortest distance from a point and a line is the perpendicular length between them, so ED and FD are the shortest lengths between D and each of the sides.

• If XXXXXX is the XXXXXX  bisector of AB, then AC = CB

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

The Angle Bisector Theorem and its converse can be rewritten as a biconditional: A point is on the angle bisector if and only if it is equidistant from the sides of the triangle.

Concurrency of Angle Bisectors Theorem: The angle bisectors of a triangle intersect in a point that is equidistant from the three sides of the triangle.

Medians

Concurrency of Medians Theorem: The medians of a triangle intersect in a point that is two-thirds of the distance from the vertices to the midpoint of the opposite side.

Altitudes

The altitude does not have to be inside the triangle. In an obtuse triangle, the altitude is outside the triangle. In these cases, you find the altitude the same way, but imagine that the opposite side extends further out and allow the altitude to be perpendicular to it. As a result, the orthocenter can be inside or outside of the triangle, depending on the triangle type.

Notes