The Midcentre, Median Centre or Centroid
The lines joining the vertices A, A', A'' to the mid-points M, M', M'' of the sides are called medians and are concurrrent at a point called the median centre (or centroid) of the triangle. [Proof: The lines A'A'' and M'M'' are parallel, and the lines A'M' and A''M'' joining their ends meet (say at G). Therefore the line MN joining their mid-points also passes through the same centre of similarity, G.].
MM'M'' is the medial triangle (or complementary triangle); its sides are parallel to those of AA'A'' and it divides it into four congruent triangles AM''M', A'MM'', A''M'M and MM'M'' similar to the whole triangle.
The centroid is the centre of mass of a uniform triangular region.
The medians divide the triangle into two triangles of equal area. They also divide the triangles GAA', GA'A'', GA''A into two equal parts. Subtraction (e.g. of GMA' from AMA' giving GAA') shows that the three triangles GAA', GA'A'', GA''A are equal in area. i.e. D/3. Thus the areal coordinates of G are (1/3, 1/3, 1/3). G is a point of trisection of the medians.
The triangle A*B*C* is ABC rotated 180° about G (* denotes reflection in G). These two triangles, forming a centrosymmetric 6-pointed star, trisect each other's sides. The lines through G parallel to the sides meet the other sides at these points of trisection, R, R*, S, S*, T, T*. These lines dissect the original triangle into 9 equal triangles.
The lines joining the vertices ABC and A*B*C* to the points of trisection form a centrosymmetric 12-pointed star-polygon. These side trisectors meet in pairs on the six lines through G, four intersections on each line. These 24 points also form a 12-ponted star, but one that is composed of three intersecting parallelograms. The outer 12 form a hexagon. The inner 12 delineate an oval shape.
Areal coordinates: LMN (0, 1/2, 1/2) and permutes. A*B*C* (–1/2, 3/4, 3/4). L*M*N* (2/3,1/6,1/6), these are the other points of trisection of the medians. L'*M'*N'* (1/2,1/4,1/4), these are the midpoints of the medians. R,R*,S,S*,T,T* (0,1/3,2/3), R',R'*,S',S'*T',T'* (1/2,1/3,1/6), these are the midpoints of the trisectors.