If I am given a triangle of any shape, and I label its edges as $e_1,e_2,e_3$, can I find two lines $L_1$ and $L_2$ that fulfill the following conditions:
$L_1$ is parallel with $e_1$,
$L_2$ is parallel with $e_2$,
$L_1$ and $L_2$ split the triangle into four different components: a triangle and three quadrilateral such that all of these components have equal area.
Do lines $L_1$ and $L_2$ like this exist? I know that without loss of generality we may assume that the triangle is equilateral since all triangles are affine equivalent.
Assume that apart from the quadrilateral at the apex $v_3$, the other 3 areas are the same.
Hint: Apply similarity to show that this apex quad has area $6 - 4 \sqrt{2} \neq 1$, hence such a configuration is not possible.