Basic Geometry: Partitions and Intersections

Question

Once more unto the breach, dear friends, once more!

So I'm currently working on a problem which I have somehow been able to simplify to the point where if I can simply prove that if three lines each partition a compact subset of $\mathbb{R}^2$ into two pairs of equal parts, then the portion of the compact set contained in the "triangle of intersections" between these lines must be zero.

To illustrate, consider this figure:

Where we know that

$$ \begin{align} A & = D_1 + D_2 \\\\ B_1 + B_2 & = C_1 + C_2 \\\\ B_2 & = C_1 + D_1 \\\\ C_2 & = B_1 + D_1 \\\\ A + C_1 & = B_2 + D_2 \\\\ A + B_1 & = C_2 + D_2 \end{align} $$

If I can just prove that $D_1 = 0$, then everything would work. Alas, after hours of banging my head against the wall (and also doing some computations, mind you), I am unable to get it to work.

Thus, I appeal to you.

Looking forward to your answers.

Answer 1

Your equations can be solved for $A,B_1, B_2$ in terms of the $C$'s and $D$'s: $$ \eqalign{A &= D_1 + D_2\cr B_1 &= C_2 - D_1\cr B_2 &= C_1 + D_1}$$ You'll want $C_2 \ge D_1$ so $B_1 \ge 0$, but otherwise $C_1, \ldots, D_2$ can be arbitrary nonnegative numbers.

Answer 2

Um, you should just have tried with a non-rectangular shape...

Clearly, the green (vertical) and blue (horizontal) lines halve the tromino, but the red (diagonal) line does not, and you would have to translate it a bit to make it halve the tromino.