Euclidean Geometry

5 Special Lines in a Triangle

Altitude, median, and the three bisectors

This article uses Similarity. Please kindly read Theorem 1.1 beforehand if you’re unfamiliar with it.

In this article and many more to come, we will discuss some unique properties of triangles. Given an arbitrary triangle, define the following:

• Altitude: a line that passes through a vertex and the opposite side, and forms a right angle with that side.
• Median: a line that passes through the midpoint of a side and the opposite vertex.
• Perpendicular bisector: a line that intersects a side of the triangle at its midpoint and makes a right angle with the side.
• Internal/interior angle bisector: a line that splits an internal angle of the triangle into two congruent parts.
• External/exterior angle bisector: a line that splits an external angle of the triangle into two congruent parts.

Since a triangle has three sides and vertices, it also has three altitudes, medians, perpendicular bisectors, internal angle bisectors, and external angle bisectors. For example, the altitude, median, perpendicular bisector, internal angle bisector, and external angle bisector corresponding to the vertex A of △ABC is shown respectively as a, b, c, d, and e in Figure 1.2. Often we call an internal angle bisector simply an angle bisector without emphasizing it is for an internal angle.