Medians, Centroid and Special Lines and Points of a Triangle
Every triangle has four classical centers: the centroid (where the medians meet), the circumcenter (perpendicular bisectors), the incenter (angle bisectors) and the orthocenter (altitudes). In brief: the centroid divides each median in the ratio 2 : 1 counting from the vertex and is the average of the three vertices, G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3). For vertices (0, 0), (6, 0) and (3, 9) it is (3, 3).
The three lines of each type always meet in one point, which is a surprising and useful fact. These points describe the “middle” of a triangle in different senses and are the centers of the circles related to the triangle. This guide defines each special line and point, shows how to compute them, gives a table with their properties and describes the Euler line.
What are medians and the centroid?
A median connects a vertex with the midpoint of the opposite side. The three medians meet at the centroid G, the center of mass of a triangular plate. The centroid is always inside the triangle and divides each median in the ratio 2 : 1, with the longer part next to the vertex. A median also divides the triangle into two triangles with equal areas.
Example 1: coordinates
A(0, 0), B(6, 0), C(3, 9). The midpoint of BC is (4.5, 4.5). The median from A goes through (0, 0) and (4.5, 4.5). The centroid is G = ((0 + 6 + 3)/3, (0 + 0 + 9)/3) = (3, 3). Check the 2 : 1 ratio: from A (0, 0) to G (3, 3) is √18, and from G to the midpoint (4.5, 4.5) is √(1.52 + 1.52) = √4.5. The ratio √18 : √4.5 = 2 : 1 ✓.
What are perpendicular bisectors and the circumcenter?
The perpendicular bisector of a side is the line perpendicular to the side through its midpoint. The three perpendicular bisectors meet at the circumcenter O, which is equally far from all three vertices. The circle with center O through the vertices is the circumcircle, with radius R. The circumcenter lies inside an acute triangle, at the midpoint of the hypotenuse in a right triangle and outside an obtuse triangle.
For a right triangle with legs 6 and 8, the hypotenuse is 10, so R = 5.
What are angle bisectors and the incenter?
The angle bisectors meet at the incenter I, which is equally far from the three sides. The circle with center I touching the sides is the incircle, with radius r = A/s, where A is the area and s is the semi-perimeter. The incenter is always inside the triangle. For the 3-4-5 triangle: A = 6, s = 6 and r = 1. The incircle touches the legs at distance 1 from the right-angle vertex.
What are altitudes and the orthocenter?
An altitude is the perpendicular from a vertex to the line containing the opposite side. The three altitudes (or their extensions) meet at the orthocenter H. It lies inside an acute triangle, at the right-angle vertex in a right triangle and outside an obtuse triangle. For vertices A(0, 0), B(6, 0), C(3, 9), the altitude from C is the line x = 3. The altitude from A is perpendicular to BC (slope −3), so its slope is 1/3 and the line is y = x/3. At x = 3: y = 1. So H = (3, 1).
What is the Euler line?
In every triangle the centroid G, the circumcenter O and the orthocenter H lie on one line, the Euler line, and G divides the segment OH in the ratio OG : GH = 1 : 2. For the triangle above, the circumcenter lies on x = 3 at the point equally far from A and C: O = (3, y) with 9 + y2 = (9 − y)2, so 9 = 81 − 18y and y = 4. So O = (3, 4), G = (3, 3) and H = (3, 1), which are collinear. The distances OG = 1 and GH = 2 confirm the ratio 1 : 2. In an equilateral triangle all four centers coincide.
Summary of the four centers
| Center | Lines | Property | Position |
|---|---|---|---|
| Centroid G | medians | divides medians 2 : 1, center of mass | always inside |
| Circumcenter O | perpendicular bisectors | equidistant from the vertices | inside, on hypotenuse or outside |
| Incenter I | angle bisectors | equidistant from the sides | always inside |
| Orthocenter H | altitudes | lies on the Euler line | inside, at right vertex or outside |
Where do these ideas come from?
Euclid’s Elements shows the concurrency of the perpendicular bisectors (through the circumscribed circle) and the angle bisectors (through the inscribed circle). The medians’ concurrency was known to Archimedes in connection with the center of gravity. The Euler line was found by Leonhard Euler in 1765, and the concurrency of altitudes was proved by various geometers. A fifth center on the same line is the center of the nine-point circle.
Where are they used today?
The centroid gives the balance point of a triangular shape: a triangular sheet balances on a pin at G. Engineers use the circumcenter for the center of a circle through three points, for example, to reconstruct a circular arc. The incenter appears in the design of the largest circle that fits in a triangular space. In computer graphics and mesh generation, all centers of triangles are used.
Which mistakes are common?
Mixing up median and altitude
A median goes to the midpoint of the side. An altitude is perpendicular to the side. They coincide only for the base of an isosceles triangle and in an equilateral triangle.
Placing the circumcenter incorrectly
In an obtuse triangle, the circumcenter lies outside, and in a right triangle it lies at the midpoint of the hypotenuse.
Wrong ratio on the median
The centroid is 2/3 of the way from the vertex, not 1/2.
Confusing the circumcircle and the incircle
The circumcircle passes through the vertices and the incircle touches the sides.