Triangle Lines
3 Problems • 3 sub-topics
Adalynn Le • 5/26/2026
Table of Contents
Introduction
Triangle lines is an ambiguous term, but in this case it refers to lines, line segments, and rays that pass through specified parts of a triangle. There are a multitude of such lines, each with their own theorems, concurrency, etc. but for simplicity we have gathered them into one encyclopedia.
select_check_box What is it?Medians
A median is a line segment that extends from a vertex (pointy corner) to the midpoint of the opposite side. I often like to think of Medians as the odd ones out because they pertain less to strict angles than any of the other lines. In isoceles triangles, the median from vertex angle to base is also the altitude, angle bisector and perpendicular bisector.
For all of the lines we will discuss in this article, the point where they meet is known as the point of concurrency. A point of concurrency is just a point where multiple lines intersect. The point of concurrency of the three medians of a triangle (and yes, they will always meet at one point) is known as the centroid. Arguably, the most important feature of a centroid is its ability to "balance" the triangle. It divides the triangle into six smaller triangles of equal area and each each median into a ratio of \(2:1\), with the longer segment being closer to the vertex.
Altitudes
An altitude is a line segment that extends from a vertex to the opposite side (or its extension) and is perpendicular to that side. For some obtuse triangles, the altitude from the obtuse angle will extend outside of the triangle. Just like medians, and you'll soon notice like every other triangle line, in isoceles or equilateral triangles, the altitude from vertex angle to base is also the median, angle bisector and perpendicular bisector. Altitudes are particularly helpful, and also likely to be the most common triangle line because they are used to find the area of a triangle. They represent the "height" or the distance from the vertex to the base, thus for area, it satisifes \(\frac{b \times h}{2}\). The point of concurrency of a the altitudes of a triangle is called the orthocenter.
Angle Bisectors
An angle bisector is a line segment or ray that divides an internal angle of a triangle into two perfectly equal angles. Every triangle has three unique angle bisectors, and they meet concurrently at a central coordinate called the incenter. The incenter is uniquely positioned because it is completely equidistant from all three outer boundaries of the triangle. Because of this property, it maps perfectly as the direct center of the triangle's inscribed circle (incircle).
Perpendicular Bisectors
A perpendicular bisector is a straight line that passes through the midpoint of a triangle's side at a perfect right angle (90°). Unlike medians or altitudes, a perpendicular bisector does not necessarily originate or pass through any opposite vertices. The trio of perpendicular bisectors intersect at a structural hub known as the circumcenter. This location represents the outer-radius origin that builds a perfect circumscribed circle (circumcircle) touching all three outer vertices.
Conclusion
The lines and rays within triangles make them exceptionally important and allow us to grap almost every important thing to know about triangles. With knowledge of those four simple concepts, you can find anything from side lengths to angles, from areas to ratios, and so on.