There are two scales: one of length $z$ and the other of $y$. these scales is placed in a row so that their ends are each supported on opposite walls of the hall and that the scales intersect. Also scales foot from the same level of corridor.
Find the height $h$ of the intersection of scales in function of the corridor has a length $x <\min (y,z) $. (h=f(x))
Let $x = a + b$ where $a$ is the length to the left of $h$ and $b$ is the length to the right of $h$
From similarity on the left triangle, we have $$ \frac{b}{h} = \frac{a+b}{y} $$ and for the triangle on the right we have $$ \frac{a}{h} = \frac{a+b}{z} $$ Dividing these two equalities, you get $$ \frac{a}{b} = \frac{y}{z} $$ From the first equality, you have $$ h = y\times \frac{b}{a+b} $$ Notice that $$ \frac{b}{a+b} = \frac{z}{y+z} $$ which should help you conclude that $$ h = \frac{yz}{y+z}. $$