Median of a triangle

The median of a triangle is a line segment that joins one of its vertices with the center of the opposite side.

Every triangle has three medians (m_a, m_b and m_c), one from each vertex. The lengths of the medians can be obtained from Apollonius’ theorem as:

Where a, b, and c are the sides of the triangle with respective medians m_a, m_b and m_c from their midpoints.

A triangle‘s three medians are always concurrent. The point where the medians intersect is the barycenter or centroid (G).

In any median of a triangle, the distance between the center of gravity (or centroid) G and the center of its corresponding side is one third (1/3) of the length of that median, i.e., the centroid is two thirds (2/3) of the distance from each vertex to the midpoint of the opposite side.

Each median divides the triangle into two triangles with equal areas.

Indeed, the two triangles Δ ABP and Δ PBC have the same base. AP = PC, by the same definition of the median, and the same altitude h referred to that line of the two bases from the vertex B.

In physics, the barycenter, or centroid (G), would be the center of gravity of the triangle.

Apollonius’ Theorem

In a triangle, it is true that the sum of the squares of two of its sides is equal to the sum of half the square of the third side and twice the square of the median corresponding to this third side.

Now, in a triangle Δ ABC, we have:

Where a, b, and c, are the legs and m_b is the median corresponding to side b.

Exercise

Find the length of the median of a triangle Δ ABC if length of sides are a=2 cm, b=4 cm and c=3 cm.

Using the equation of Apollonius’ theorem, we can find the length of all three medians:

Thus, the medians are m_a=3.39 cm, m_b=1.58 cm and m_c=2.78 cm.

Lines Associated with a Triangle

• Altitude of a Triangle
• Angle Bisector of a Triangle
• Perpendicular Bisector of a Triangle

Centroid of a Triangle

The centroid of a triangle (or barycenter of a triangle) G is the point where the three medians of the triangle meet.

The medians of a triangle are the line segments created by joining one vertex to the midpoint of the opposite side. Since every triangle has three sides and three angles, it has three medians (m_a, m_b and m_c).

Centroid theorem: the distance between the centroid and its corresponding vertex is twice the distance between the barycenter and the midpoint of the opposite side. That is, the distance from the centroid to each vertex is 2/3 the length of each median. This is true for every triangle.

The centroid is always inside the triangle.