Proof that the three medians of a triangle meet at one point, known as the centroid. Proof 3: Proof by vectors

This is the vector a.This is the vector b. a b We may add vectors by placing them end-to-end. a + b = (a + b) a + b b a a + b = (a + b) b + a = (a + b) The commutative law for addition holds for vector algebra!

a b a + b We will assume that our common laws for algebra hold for vector algebra, and try to prove our theorem. After testing our theorem on real world triangles, then proving it twice, we are sure that it is true. So we know that our new technique should lead to a proof.

a b a + b Once again, we need to prove that each of the three medians of any triangle meet at one point, known as the centroid, and that this point cuts each median in a ratio of 2:1. Our strategy will be to find the vector for each median, then find 2/3 of these vectors, then verify that the vectors are identical.

a b a + b The first vector is easy. a + ½ b If we add half of vector b to vector a, we get vector (a + ½ b), our median. The vector that goes 2/3 of the way to (a + ½ b) is We may write this as: Let’s remember that. Centroid #1

a b a + b Centroid #1 Our next median is trickier, because it doesn’t start at the origin. Vectors, however, do start at the origin. So, we need to find the vector that runs parallel to our next median, find 1/3 of that, then shift it away from the origin. ½ a It will help to draw in some extra lines.The vector that is the same direction and magnitude as our median is b + ½ a. Our centroid should be located at ½ a + 1/3(b + 1/2a) 1/3(b + 1/2a) 1/2a But this is equal to (2a+b)/3 ! Centroid #2 b ½ a

a b a + b Centroid #1Centroid #2 Here’s our next median. Once again, we will need to add some lines to identify a similar vector. b a -a-a a + b It looks like our vector is –a + ½ (a + b). That is equal to ½ (b – a). We take 2/3 of ½ (b – a), which equals 1/3 (b – a). Finally, we shift that back where it belongs, by adding a to it. a +1/3 (b – a) = (2a + b)/ 3 Centroid #3

a b a + b Centroid #1Centroid #2Centroid #3 The theorem is proved. All three medians intersect at the vector (2a + b) / 3.