When will the ratio of the areas of the two triangles equal 4?

When triangle ABC is P is the pt of concurrency of the medians, the centroid, the ratio of the areas of the two triangles will be 4.

This is true because, since P is the centroid, F, D, and E are midpoints. This makes Triangle FDE the same size as Triangles AFE, FBD, and DEC. So, Triangle FDE makes up 1/4 the area of Triangle ABC.

Can there ever be a triangle DEF that is less than 1/4 the area of ABC?

Click here to open a GSP sketch to manipulate point P, and consider the ratio of the areas.

We can see in the GSP sketch that triangle ABC will always be at least 4 times larger than triangle DEF.

Notice that the farther the P gets from the centroid, that the larger the ratio of ABC to DEF gets.

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