Geometry

Bisectors, Medians, Altitudes

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

Midsegment: The segment that joins the midpoints of a pair of sides of a triangle.

Perpendicular Bisector: A line, ray, or segment that passes through the midpoint of a segment and intersects that segment at a right angle.

Equidistant: The same distance from one figure as from another figure.

Median: A line segment drawn from one vertex of a triangle to the midpoint of the opposite side.

Altitude: A line segment drawn from a vertex of the triangle and is perpendicular to the other side.

Point of Concurrency: The point where three or more lines intersect.

Circumcenter: The point of concurrency for the perpendicular bisectors of the sides of a triangle.

Incenter: The point of concurrency for the angle bisectors of a triangle.

Centroid: The point of concurrency for the medians of a triangle.

Orthocenter: The point of concurrency for the altitudes of a triangle.

Slope of a Line

For every triangle, there are three midsegments.

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

Slope of a Line

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 \(\color{red}{\overleftrightarrow{CD}}\) is the  \(\perp\) bisector of AB, then AC = CB
  • If XXXXXX is the XXXXXX  bisector of AB, then AC = CB
Perpendicular Bisectors

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.
    Isosceles Triangles

    This is not true for any angle other than the vertex angle!

    Geometry

    Bisectors, Medians, Altitudes Cont.

    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.

    Concurrency of Perpendicular Bisectors
    • If \(\color{green}{\overline{PC}}\), \(\color{green}{\overline{QC}}\), and \(\color{green}{\overline{RC}}\) are \(\perp\) bisectors, then \(\color{blue}{LC}\) = \(\color{blue}{MC}\) = \(\color{blue}{OC}\)
    circumcenter
    • Perpendicular bisectors intersect at the circumcenter, and a circumcircle can be drawn around the triangle with its center at the circumcenter.

    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
    Angle Bisectors

    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.

      • Angle bisectors meet in the incenter.
      • A circle can be inscribed in any triangle with its center at the incenter
      Angle Bisectors

      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.

        • \(AG = \frac{2}{3}AD, EG = \frac{2}{3}BE, CG = \frac{2}{3}CF\)
        • The centroid is the balancing point of the triangle
        Medians
        centroid is the balancing point

        Geometry

        Bisectors, Medians, Altitudes Cont.

        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.

        Acute Triangle

        Right Triangle

        Obtuse Triangle

        Acute Triangle
        Right Triangle
        Obtuse Triangle

        The orthocenter is inside the triangle.

        The legs of the triangle are two of the altitudes.

        The orthocenter is the vertex of the right angle.

        The orthocenter is outside the triangle.

        Here is a way to remember the different points of concurrency. Remember the first letter of each word in this saying:     The first letters correspond to:
        Peanut Butter              CookiesPerpendicular Bisectors Circumcenter
        Are Best In                 Angle Bisectors Incenter
        Milk Chocolate           Medians Centroid
        And Ovaltine              Altitudes Orthocenter

        Notes