Crossed ladders problem revisited

     

Find the area of ​​the figure denoted by ?, knowing the areas of the other colored triangles X, Y, Z.

 ANSWER CiP

$$\textbf{S}_?=\frac{X\cdot Z \cdot(X+2Y+Z)}{Y^2-X\cdot Z}$$

                         Solution CiP

          It is just a takeover from other sources, but the calculations are mine.

          From the web-page Crossed ladders problem , we find a formula

 the formula links several areas together. In our notation

$$\frac{1}{\textbf{S}_{AEC}}+\frac{1}{\textbf{S}_{ADC}}=\frac{1}{\textbf{S}_{AFC}}+\frac{1}{\textbf{S}_{ABC}}.$$

          Substituting above with our data we have

$\frac{1}{X+Y}+\frac{1}{Y+Z}=\frac{1}{Y}+\frac{1}{\textbf{S}_?+X+Y+Z}$

so  $\frac{1}{\textbf{S}_?+X+Y+Z}=\frac{1}{X+Y}+\frac{1}{Y+Z}-\frac{1}{Y}=\frac{Y(Y+Z)+Y(X+Y)-(X+Y)(Y+Z)}{Y(X+Y)(Y+Z)}=\frac{Y^2-XZ}{Y(Y+X)(Y+Z)}.$

Hence $\textbf{S}_?=\frac{Y(Y+X)(Y+Z)}{Y^2-XZ}-X-Y-Z$ and we only get the answer by doing some calculations.

$\blacksquare$