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:
$\mathrm{\Delta }$ABC and $\mathrm{\Delta }$AFE are similar so ${\displaystyle \frac{z}{x}}={\displaystyle \frac{b}{a+b}}$. -- equation 1
Cross multiply and you have z ( a + b ) = bx

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

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

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

Applicable question: 

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

$\overline{AB}={\displaystyle \frac{20\times 30}{(20+30)}}$ = 12