As I showed before, the area of a triangle is 0.5 (a*d - b*c) where u = (a, b) and v = (c, d) are two vectors that are the triangle's sides. To prove the really surprising fact about the centroid, all that must be shown is that . The vector v is the side, so we need its magnitude; the perpendicular vector is adjusted by the scale factor g so that it is the proper distance from the side v to the centroid.
Finding g takes a little more work. Again as shown before, the vector w to the centroid is (u+v)/3, that is (a+c, b+d)/3. Using the Pythagorean theorem, it is true that . One also knows from vector addition that . This allows a substitution for fv: . Therefore, substituting into the Pythagorean theorem yields:
Therefore, our proof is nearly done. Going back to the original statement to prove,
And our work is done.