, 2022

Solution:

Let $P_{i}(x, y, z)$ be the point with barycentric coordinates $(x, y, z)$ in triangle $A_{i} B_{i} C_{i}$. Note that since this is linear in $x, y$, and $z$, the signed area of triangle $P_{1}(x, y, z) P_{2}(x, y, z) P_{3}(x, y, z)$ is a homogenous quadratic polynomial in $x, y$, and $z$; call it $f(x, y, z)$.
We now claim that
$$f\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)=\frac{4 f\left(\frac{1}{2}, \frac{1}{2}, 0\right)+4 f\left(\frac{1}{2}, 0, \frac{1}{2}\right)+4 f\left(0, \frac{1}{2}, \frac{1}{2}\right)-f(1,0,0)-f(0,1,0)-f(0,0,1)}{9}$$
This is easy to verify for $f \in\{x^{2}, y^{2}, z^{2}, x y, x z, y z\}$, after which the statement follows for general $f$ by linearity. Then, assuming that we can arbitrarily choose the signs of the areas, the area is maximized at
$$\frac{4 \cdot 2061+9}{9}=229 \cdot 4+1=917$$
Now it remains to show that this best-case scenario is actually possible. The first step is to first show that these values from an actual $f$, i.e. that one can fit a homogenous quadratic polynomial through every six possible values for $f$ at the six given points. One way to see this is to note that by choosing the coefficients for $x^{2}, y^{2}$, and $z^{2}$, the values at the vertices of the triangle can be matched, while adding any of the $x y, x z$, and $y z$ terms influences only one of the midpoints, so they can be matched as well.
Now we show that this particular $f$ can be realized by a choice of triangles. To do this, note that by continuity there must exist $x_{0}, y_{0}$, and $z_{0}$ with $f\left(x_{0}, y_{0}, z_{0}\right)=0$, since $f(1,0,0)$ and $f\left(\frac{1}{2}, \frac{1}{2}, 0\right)$ are different signs, and introduce the new coordinates $u=x-x_{0}$ and $v=y-y_{0}$; then $f$ can be written as $a u^{2}+b u v+c v^{2}+d u+e v$. Now, one can let $P_{1}(u, v)=(0,0)$, $P_{2}(u, v)=(u, v)$, and $P_{3}(u, v)=(-c v-e, a u+b v+d)$. This can be shown to reproduce the desired $f$.
Finally, to address the condition that the original triangles must be nondegenerate, we can perturb each of the $P_{i}$ by a constant, which doesn't affect $f$ as areas are translation-invariant. This concludes the proof.