Three Pole Problems : Similar triangles

Question: If you know the length of x and y, and the whole length of $\overline {AB}$,

A: what is the ratio of a to b and 

B: what is the length of z.






 Solution for question A:
$\Delta$ABC and $\Delta$AFE are similar so $\dfrac {z} {x}=\dfrac {b} {a+b}$. -- equation 1
Cross multiply and you have z ( a + b ) = bx

$\Delta$BAD and $\Delta$BFE are similar so $\dfrac {z} {y}=\dfrac {a} {a+b}$. -- equation 2
 Cross multiply and you have z ( a + b ) = ay

bx = ay so $\dfrac {x} {y}=\dfrac {a} {b}$  same ratio

Solution for question B: 
Continue with the previous two equations, if you add equation 1 and equation 2, you have:
$\dfrac {z} {x}+\dfrac {z} {y}=\dfrac {b} {a+b}+\dfrac {a} {a+b}$
$\dfrac {zy+zx } {xy}=1$ $\rightarrow$ z = $\dfrac {xy} {x+y}$

Applicable question: 

$\overline {CD}=15$ and you know $\overline {DB}:\overline {BC}=20:30=2:3$ 
 so  $\overline {DB}=6$ and $\overline {BC}=9$ 

$\overline {AB}=\dfrac {20\times 30} {\left( 20+30\right) }$ = 12