Areas and Centroid in a Triangle

The applet may suggest the following statement:

From a point O inside ABC draw the lines OL, OM, ON parallel to the sides BC, AC, and AB, respectively so that, L lies on AB, M, on BC, and N on AC. It so happens that the areas of triangles BOL, COM, AON are equal. Prove that O is the centroid of ABC.

Proof

(For the proof, drag the inner point to coincide with the "target" point inside the triangle and press the "Hint" button.)

Let OL intersect AC in L'. Then

such that

But since LL'||BC, the altitudes in triangles BOL and COL' (from B and C, respectively) are equal. Therefore,

which implies that O lies on the median from vertex A. Similarly, it lies on the medians from B and C. Thus O is none other than the centroid of ABC.

References