5.3 Bisectors, medians, and altitudes

Construction of Triangle Bisectors and Lines

Every triangle has four families of special lines you can draw: angle bisectors, perpendicular bisectors, medians, and altitudes. Each one is constructed differently and reveals different information about the triangle.

Angle bisector — A ray that divides an angle of the triangle into two congruent angles. It extends from a vertex into the interior of the triangle. Note that the angle bisector does not generally pass through the midpoint of the opposite side (that only happens in special cases like an equilateral triangle).

Perpendicular bisector — A line that passes through the midpoint of a side and is perpendicular to that side. Unlike the other three constructions, a perpendicular bisector doesn't have to pass through a vertex at all.

Median — A segment connecting a vertex to the midpoint of the opposite side. Every triangle has three medians, and each one splits the triangle into two smaller triangles of equal area.

Altitude — A segment from a vertex that meets the opposite side (or its extension) at a right angle. The side it meets is called the base for that altitude. In an obtuse triangle, two of the altitudes fall outside the triangle, so you extend the opposite side to find the foot of the altitude.

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Properties of Triangle Bisectors and Lines

Each type of line carries its own set of useful properties.

• Angle bisectors — The angle bisector from a vertex divides the opposite side into two segments whose lengths are proportional to the two adjacent sides (this is the Angle Bisector Theorem, detailed below). All three angle bisectors meet at a single point called the incenter, which is equidistant from all three sides of the triangle.
• Perpendicular bisectors — Any point on the perpendicular bisector of a segment is equidistant from the segment's two endpoints. Because of this, all three perpendicular bisectors of a triangle meet at the circumcenter, which is equidistant from all three vertices.
• Medians — The three medians meet at the centroid, which divides each median in a $2:1$ ratio, with the longer portion between the vertex and the centroid. The centroid is also the triangle's center of gravity (balance point).
• Altitudes — The three altitudes meet at the orthocenter. Its position depends on the triangle type:
  • Acute triangle — orthocenter lies inside the triangle
  • Right triangle — orthocenter lies on the vertex of the right angle
  • Obtuse triangle — orthocenter lies outside the triangle

Points of Concurrency in Triangles

A point of concurrency is where three or more lines intersect at a single point. Each family of triangle lines produces exactly one.

A few things worth remembering:

• The incenter always lies inside the triangle, regardless of triangle type.
• The circumcenter lies inside acute triangles, on the hypotenuse of right triangles, and outside obtuse triangles.
• The centroid always lies inside the triangle.
• The orthocenter can be inside, on, or outside the triangle (see altitude properties above).

Relationships Between Triangle Lines

Angle Bisector Theorem

If $\overline{AD}$ bisects $\angle BAC$ in $\triangle ABC$, with $D$ on $\overline{BC}$, then:

$\frac{BD}{DC} = \frac{AB}{AC}$

The two segments of the opposite side are proportional to the two sides of the triangle that form the bisected angle. This is especially useful for finding unknown side lengths when you know the bisector's position.

Centroid and Median Ratios

The centroid $G$ sits exactly $\frac{2}{3}$ of the way from each vertex to the midpoint of the opposite side. So if a median has total length 12, the distance from the vertex to the centroid is 8, and from the centroid to the midpoint is 4.

Euler's Line

Three of the four concurrency points are always collinear (they lie on the same line), called Euler's line. Specifically, the circumcenter $O$, centroid $G$, and orthocenter $H$ satisfy:

$OG : GH = 1 : 2$

The centroid sits one-third of the way from the circumcenter to the orthocenter. The incenter generally does not lie on Euler's line (the one exception is the equilateral triangle, where all four points coincide at the same location).