# The Medians of a Triangle are Concurrent

The medians of any triangle are concurrent and that the point of concurrency divides each one of them in the ratio **2:1**.

Consider the triangle as shown in the diagram and suppose that , and are the vertices of the given triangle . As we know, the median is defined as the line segment joining the vertex of the triangle to the midpoint to the opposite side of the triangle.

Now suppose that , and are the midpoints of the sides of the triangle , so these midpoints can be calculated using the midpoint formula as follows:

The midpoint of side is

The midpoint of side is

The midpoint of side is

Consider is the point intersection of the three medians of the triangle .

First we find the coordinates of the point with respect to the median , since the point divides the median in the ratio , i.e. . We find the point using the division formula or ratio formula (internal division).

Let and

Second we find the coordinates of the point with respect to the median , since the point divides the median in the ratio , i.e. . We find the point using the division formula or ratio formula (internal division).

Let and

Similarly, the coordinates of with respect to .

Hence, the medians of the triangle are concurrent and the point of concurrency is