Center of a Triangle - The Centroid - The Intersection Point of Medians

The center point of a triangle is also called the intersection point of medians or the meeting point of medians.
Remember - a median is a line segment that extends from a vertex to the opposite side and divides it exactly in half.
Let's see this in the illustration:

In triangle $ABC$ shown here, we can see that the purple point $M$ represents the intersection point of the three medians in the triangle.
Point $M$ is also the centroid of the triangle.
Important theorems about the intersection point and the medians in a triangle:

The theorem states that if $2$ medians intersect at a certain point, then the third median in the triangle must also pass through the same point and intersect at that point, which is called the centroid.

Let's look at an example:

In triangle $ABC$ there are two medians $AD$ and $BE$ intersecting at point $M$.
From this, it follows that if segment $CW$ is a median, it must pass through point $M$, and conversely, if $CE$ passes through point $M$, we can determine that it is a median to side $AB$
Note: We can determine that if $2$ medians in a triangle intersect at a certain point, it will be the centroid.

Let's practice the first theorem about the centroid:
Here is triangle $ABC$

Given that:
$CE$ is a median in the triangle
$BW$ is a median in the triangle
and - $AD$ passes through point $M$.

It is also known that:

$DB=5$
$BE=4$
$​​​​​​​AW=4$

1. Find $CD$
2. Find the perimeter of the triangle

Solution:

1. We know that $AD$ passes through point $M$ which is the same point where the two medians $CE$ and $BW$ intersect.
   Therefore, according to the theorem that all three medians intersect at one point, we can determine that $AD$ is also a median because if $2$ medians meet at a certain point, the third median must pass through it as well.
   We are given that $DB=5$ therefore $CD=5$ because a median divides the side into two equal parts.
   
2. To find the perimeter of the triangle we need to find all sides.

$AE = 4$ since $CE$ is a median
$CW = 4$ since $BW$ is a median

And we found $CD$ in part a.
Therefore:
$AB+BC+AC=$
$8+10+8=26$

The perimeter of triangle $ABC$ is $26$ cm.

Let's look at an example:

In triangle $ABC$ the three medians intersect at point $M$.
According to the theorem, point $M$ divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex.
Thus we can determine that:
$AM=2x$
$MD=x$

And:
$CM=2Y$
$ME=Y$

And:
$BM=2Z$
$MW=Z$

Now we will practice the second theorem about the centroid:
Here is triangle $ABC$

Given that:
$AD$ is a median
$BW$ is a median
and $CE$ passes through point $M$

It is also given that: $ME=2$
and $BM=5$

Find $CM$ and $WM$
Solution:
Since we are given that: $AD$ is a median and $BW$ is a median and $CE$ passes through point $M$, we can conclude that $CE$ is a median because if two medians intersect at a certain point, the third median must pass through it.
According to the second theorem which states that the intersection point of the medians divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex, and given that: $ME=2$ (the smaller part), we can conclude that:
$CM= 4$
Since $BM=5$ is the larger part closer to the vertex, we can conclude that $WM=2.5$