Centroids of Triangles
First, the question that begs to be asked, what is a centroid? A centroid of the circle is defined as the intersection of all three medians of a triangle.
What would happen if we used an obtuse triangle rather than an acute triangle?
You may notice that the centroid is still in approximately the same location- the middle. Furthermore, the centroid is still inside the triangle. We can also predict that the centroid will remain inside the triangle for an equilateral triangle.
Equilateral triangles are of special interest because the centroid is in the same place as all of the other types of centers: the orthocenter, the circumcenter, and the incenter.
We can also easily show that the centroid divides each median into a ratio of 2:1 with an equilateral triangle.
We can also see the medians divide the equilateral triangle into three congruent triangles.
More specifically, the apothem bisects the triangle into three 30-60-90 triangles. As we know,
Based on these congruent triangles, we know the length of BD is also 2. Now, we have our ratio of 2:1.
Perhaps we should also consider isosceles triangles.
The centroid again remains within the boundary of the sides of the triangle.
Why are centroids important? Centroids are most useful for studying centers of gravity and moments of inertia in physics and engineering. So, it seems logical that the centroid should remain within the triangle; only irregular shapes with extended sides have centers of gravity on the exterior.