Median of a triangle

A median of a triangle is a line segment joining a vertex of the triangle to the midpoint of the opposite side. A triangle has exactly three medians, and the point where they meet is the centroid of the triangle.

Example

Look at the following triangle with its medians marked: The medians are marked in red. As with the altitudes of a triangle, we label the medians with a lowercase letter, here m, and a subscript that shows which side and which vertex the median belongs to. Since the side opposite vertex A is a, the median from A is called m_a.

A median cuts the triangle into two triangles with the same area. Unlike the altitudes of a triangle, the medians look the same in every type of triangle.

Midpoint of a line segment

The three points M_a, M_b and M_c are the midpoints of the sides. You can find a midpoint using a pair of compasses and a ruler. For example, to find the midpoint of side AB, draw two circles of the same radius, with centres at vertices A and B. The radius must be greater than half the length of AB – but it must be the same for both circles! Either estimate it, or draw a circle whose radius is the length of side AB. The two circles meet at two points. Join these points with a line segment, and where this segment crosses side AB is the midpoint of side AB.

The centroid of the triangle is highlighted in green in the figure. It is the “centre” of the triangle, in a sense.

Centroid

All the medians always meet at a single point. That is how you can tell whether you have drawn them correctly. If they meet somewhere else, your drawing was either inaccurate or just plain wrong. The centroid always lies inside the triangle, unlike the orthocentre, which can also lie outside it.

The centroid is, in a sense, the centre of the triangle: if you wanted to balance a triangle on the tip of a pencil, you would put the pencil exactly under the centroid so that the triangle did not fall off.

The centroid also divides each median in the ratio 1:2. That means two thirds of the length of the median lie on one side of the centroid and the remaining third on the other side. The longer part of the median is always towards the vertex of the triangle, and the shorter part is on the side towards the midpoint of the opposite side. Look at the figure:

The length of the segment AG (the blue segment) is twice the length of the segment GM_a (the green segment). The segments AD, DG and GM_a are all the same length. The same holds for the other medians.

Midline

A midline is a line segment whose endpoints are the midpoints of two sides of the triangle. Take a look at the figure:

The points M_a, M_b and M_c are the midpoints of the sides, just as for the medians. To make a midline, we simply join two of these points with a line segment. We denote the midlines by the letter s with a subscript showing the opposite vertex. The triangle that is formed is called the medial triangle. An interesting fact is that the centroid of this medial triangle is the same as the centroid of the original triangle. In our case it is the point G.

Length of a median

To work out the length of a median we can use Apollonius’ theorem. Suppose that a, b and c are the lengths of the corresponding sides of the triangle, and similarly m_a, m_b and m_c are the lengths of the medians:

$$\begin{array}{rcl}{m}_a& =& \sqrt{\frac{2b^2+2c^2-a^2}{4}}\\ m_b& =& \sqrt{\frac{2a^2+2c^2-b^2}{4}}\\ m_c& =& \sqrt{\frac{2a^2+2b^2-c^2}{4}}\end{array}$$