Trigonometry/Circles and Triangles/Area bisectors

Area bisectors

As already noted, the three medians of a triangle each bisect its area. So do the lines parallel to each side of the triangle dividing the other two sides in the ratio $$\sqrt 2+1:1$$. These six lines form four concurrent triplets: the three medians and each median with the lines parallel with the two sides not cut by the median. The latter three triplets define a triangle of which the medians are the same as those of the original triangle.