Wednesday, July 2, 2014

Centroids and area

There is a really surprising fact about centroids, one I had not known about until this year. The area of a triangle is equal to 1.5 times the distance of centroid to a side times the length of the side. It is not too hard to prove this result.

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 1.5·gv·v=0.5·ad-bc. 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 w2=gv2+fv2. One also knows from vector addition that fv=w+gv. This allows a substitution for fv: fv=a+c,b+d/3+gd, -c. Therefore, substituting into the Pythagorean theorem yields:

w2=gv2+fv2a+c29+b+d29=g2·c2+d2+a+c,b+d3+gd, -c20=2g2·c2+d2+2ga+c·d3+-2gb+dc3

This sets up showing that g = 0 (the trivial solution) or that:


Therefore, our proof is nearly done. Going back to the original statement to prove,


And our work is done.

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