Geometry and statics

The rules of statics may be applied to plane geometry.
See balance below:

Example 1

The medians in a triangle divide each other in a ratio of 2:1, counted from the angle.

Apply a force of 1 in A.
The force in C also is 1, because of the medians.
So, CS.1 = ST.2.

This is a special case of a general rule.

Example 2

A problem:

Which part of the triangle is the colored area?

Because of the symmetry we notice:

Which part of the traingle is area B?
We apply the rule:
    If triangles share an angle, their areas have the ratio of the product of the edges around the angle.
In picture:

The proof is left to the reader.

Back to the problem.

So, the yellow area = (3/7).(1/3).[CAE] = (1/7).[CAE].
[CAE] = (1/3).[CAB].

Purple colored triangle = 3.[CDS] = (1/7) of triangle ABC.