Question 166
Fix an equilateral triangle $ABC$ with side length $1$. We call $(\triangle DEF, \triangle XYZ)$ a good triangle pair if the points $D$, $E$, and $F$ lie in the interior of segments $BC$, $CA$, and $AB$, respectively, the points $X$, $Y$, and $Z$ lie on the lines $BC$, $CA$, and $AB$, respectively, and they satisfy the conditions
$$
\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}, \quad \text{and} \quad DE \perp XY, EF \perp YZ, FD \perp ZX.
$$
As $(\triangle DEF, \triangle XYZ)$ runs through all good triangle pairs, determine all possible values of $\frac{1}{S_{\triangle DEF}} + \frac{1}{S_{\triangle XYZ}}$.
$$
\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}, \quad \text{and} \quad DE \perp XY, EF \perp YZ, FD \perp ZX.
$$
As $(\triangle DEF, \triangle XYZ)$ runs through all good triangle pairs, determine all possible values of $\frac{1}{S_{\triangle DEF}} + \frac{1}{S_{\triangle XYZ}}$.