Question 31
The cities $A$, $B$ and $C$ are located at the vertices of an equilateral triangle with side $60\ \mathrm{km}$. Between them, connecting them two by two, there are three highways, $AB$, $AC$ and $BC$, all in straight lines. A railway will be built, also in a straight line, which must intersect $AB$ at $30\ \mathrm{km}$ from $A$; $AC$ at $20\ \mathrm{km}$ from $C$; and, finally, the highway $BC$ beyond $C$.
a) If $h_1$, $h_2$ and $h_3$ are the distances from $A$, $B$ and $C$ to the railway, respectively, verify that
$$
\frac{AD}{BD} = \frac{h_1}{h_2}, \quad \frac{BF}{CF} = \frac{h_2}{h_3}, \quad \text{and} \quad \frac{CE}{AE} = \frac{h_3}{h_1}
$$
b) What is the distance between the city $C$ and the intersection of the railway with the highway $BC$?
a) If $h_1$, $h_2$ and $h_3$ are the distances from $A$, $B$ and $C$ to the railway, respectively, verify that
$$
\frac{AD}{BD} = \frac{h_1}{h_2}, \quad \frac{BF}{CF} = \frac{h_2}{h_3}, \quad \text{and} \quad \frac{CE}{AE} = \frac{h_3}{h_1}
$$
b) What is the distance between the city $C$ and the intersection of the railway with the highway $BC$?