Question 21
In triangle $ABC$, let the parabola with focus $A$ and directrix $BC$ intersect sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Similarly, let the parabola with focus $B$ and directrix $CA$ intersect sides $BC$ and $BA$ at $B_1$ and $B_2$, respectively. Finally, let the parabola with focus $C$ and directrix $AB$ intersect sides $CA$ and $CB$ at $C_1$ and $C_2$, respectively.
If triangle $ABC$ has sides of length $5$, $12$, and $13$, find the area of the triangle determined by lines $A_1C_2$, $B_1A_2$, and $C_1B_2$.
If triangle $ABC$ has sides of length $5$, $12$, and $13$, find the area of the triangle determined by lines $A_1C_2$, $B_1A_2$, and $C_1B_2$.