4.12*. In the triangle $T_{a}=\triangle A_{1} A_{2} A_{3}$, the triangle $T_{b}=$ $=\triangle B_{1} B_{2} B_{3}$ is inscribed, and in the triangle $T_{b}$, the triangle $T_{c}=\triangle C_{1} C_{2} C_{3}$ is inscribed, with the sides of triangles $T_{a}$ and $T_{c}$ being parallel. Express the area of triangle $T_{b}$ in terms of the areas of triangles $T_{a}$ and $T_{c}$.
4.12. Let the areas of triangles $T_{a}, T_{b}$, and $T_{c}$ be $a, b$, and $c$. Triangles $T_{a}$ and $T_{c}$ are homothetic, so the lines connecting their corresponding vertices intersect at one point $O$. The coefficient $k$ of similarity of these triangles is $\sqrt{a / c}$. Clearly, $S_{A_{1} B_{3} O}: S_{C_{1} B_{3} O}=A_{1} O: C_{1} O=k$. Writing similar equalities and adding them, we get $a: b=k$, hence $b=\sqrt{a c}$.