Apollonius' theorem

Apollonius' theorem is a special rule that helps us learn more about triangles.

First, imagine a triangle, which is a shape with three sides. Let's call the sides A, B, and C.

Now, Apollonius' theorem says that if we draw a line through the middle of side A and then another line through the middle of side B, the place where those lines meet will be a specific point. This point is called the centroid.

The theorem also says that the distance between the centroid and the point where the line through side A meets side C will be twice as long as the distance between the point where the line through side B meets side C.

This might sound a little confusing, so let's use an example.

Imagine a triangle with sides A, B, and C.

We draw a line through the middle of side A (let's call the point where it meets A "D") and another line through the middle of side B (let's call the point where it meets B "E").

These lines cross at the centroid, which is a point in the middle of the triangle.

Now, if we draw a line from the centroid to the point where line AD meets side C (let's call this point "F"), the length of that line will be twice as long as the length of the line from the centroid to the point where line BE meets side C (let's call this point "G").

Basically, Apollonius' theorem helps us understand the relationship between the centroid and the sides of a triangle.