The centroid of a triangle is the intersection of the medians.
The medians are the line segments connecting each vertices to the midpoint of the opposite side.
Theorem : The medians of a triangle are concurrent.
Concurrent means the three medians meet at a point.
Proof : Let's just do the equations for the medians, and see where the pairs meet.
Let's use parametric forms, which are easy to write.
Median 1 endpoints (a,b) and ( {c+e}/2, {d+f}/2 ) is line segment
(x,y) = (1-t)(a,b) + t( {c+e}/2, {d+f}/2 ) quad for 0 \le t le 1.
Median 2: (c,d) and ( {a+e}/2, {b+f}/2 )
(x,y) = (1-u)(c,d) + u( {a+e}/2, {b+f}/2 ) quad for 0 \le u le 1.
Median 3: (e,f) and ( {a+c}/2, {b+d}/2 )
(x,y) = (1-v)(e,f) + v( {a+c}/2, {b+d}/2 ) quad for 0 \le v le 1.
Medians 1 and 2 meet when
(1-t)(a,b) + t( {c+e}/2, {d+f}/2 ) = (1-u)(c,d) + u( {a+e}/2, {b+f}/2 )
2(1-t)(a,b) + t( c+e, d+f ) = 2(1-u)(c,d) + u( a+e, b+f )
2(a,b) + t(c+e-2a,d+f-2b) = 2(c,d) + u(a+e-2c,b+f-2d)
That's two equations,
2a + t(c+e -2a ) = 2c+ u(a+e - 2c)
2b + t(d+f -2b ) = 2d+ u(b+f - 2d)
We solve for t,
t(c+e -2a )(b+f - 2d) = (2c-2a) (b+f - 2d)+ u(a+e - 2c)(b+f - 2d)
t(d+f -2b )(a+e - 2c) = (2d-2b)(a+e - 2c) + u(b+f - 2d)(a+e - 2c)
t ((c+e -2a )(b+f - 2d) - (d+f -2b )(a+e - 2c) ) = 2(c-a) (b+f - 2d) - 2(d-b)(a+e - 2c)
t ( ( bc + be - 2ab +cf +ef -2af -2cd -2de +4ad) - (ad + af - 2 ab +de + df -2be -2cd -2cf + 4bc) ) = 2 (bc + cf -2cd - ab -af +2ad -( ad + de -2cd -ab -be +2bc) )
t ( - 3 bc +3 be + 3 cf -3af -3de +3ad) = 2 ( - bc + cf -af +ad -de+ be -bc) )
3t ( - bc + be + cf - af - de + ad) = 2 ( - bc + cf -af +ad -de+ be -bc) )
t = 2/3
Wow, Just by symmetry we'll get u=2/3 and v=2/3 as well.
That tells us the centroid is
(1-2/3)(a,b) + 2/3( {c+e}/2, {d+f}/2 ) = ( {a+c+e}/3, {b+d+f}/3 )
We'd of course get the same answer had we used the first and third or second and third median equations.
Not only are the medians concurrent, meeting at a common point called the centroid, but we showed the centroid divides each median in the ratio 2:1, the longer piece to the vertex.
