Centers of a Triangle - Centroid
by Emily Kennedy

The centroid is defined as the point at which the three medians of a triangle meet. Let's prove that the three medians of any triangle really do meet in a single point.

First, let's find equations for the medians of a triangle.

Using the labeling system we have already described, and the midpoint formula, we can find the medians of a triangle:

is the line through the points and .

If is vertical (i.e., if a = 2b), then is the line x = a.

If is not vertical (i.e., if a ≠ 2b), then is the line described by:

or, more simply,

is the line through the points and .

If is vertical (i.e., if 2a = b), then is the line x = 2a.

If is not vertical (i.e., if 2a ≠ b), then is the line described by:

or, more simply,

is the line through the points and .

Note that cannot be vertical, since a ≥ 0 and b > 0 (so a + b > 0).

Thus, is the line described by:

or, more simply,

Let's find the point at which and intersect, and then we can determine whether lies on as well. If it does, we will have shown that all three medians intersect in a single point.

Case 1: is vertical (i.e., a = 2b)

Then we must have
= a
and

Note that and can't both be vertical
(If they were, then we would have a = 2b and 2a = b,
which implies that a = b = 0,
which contradicts the assumption that a > 0).

So in Case 1, we must have that is described by

Does our point lie on ? Let's find out.

The y-coordinate on when x = is:

So does indeed lie on .

Thus, in Case 1, the three medians all intersect at a single point.
That point is:

Case 2: is not vertical (i.e., a ≠ 2b)

We need to find a point that is on both and

The y-coordinate of both lines must be the same at x = .

So we must have:

And thus, we must have:

Does our point lie on ? Let's find out.

Case 2a: is vertical (i.e., 2a = b)

Then any point on the line has x-coordinate b.

Since 2a = b in Case 2a, we have:

Thus, in Case 2a, the three medians all intersect at a single point.
That point is:

Case 2b: is not vertical (i.e., 2a ≠ b)

Then is described by:

The y-coordinate on when x = is:

So does indeed lie on .

Thus, in Case 2b, the three medians all intersect at a single point.
That point is:

So for any triangle, all three medians intersect at a single point, the centroid. For a triangle whose vertices are (0,0), (2a,0), and (2b,2c) the coordinates of the centroid are

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