<s>[INST] Understanding a Symmetry Argument 
A rectangle has dimensions $a$ units by $b$ units with $a > b$. A diagonal divides the rectangle into two triangles. A square, with sides parallel to those of the rectangle, is inscribed in each triangle. Find the distance between the vertices (of the squares) that lie in the interior of the rectangle.

In the solution, we find that one square has an 'interior' vertex $(\frac{ab}{a+b},\frac{ab}{a+b})$, because the vertex must lie on both $y=x$ and $y = -\frac{b}{a}x + b$. Then the author asserts that symmetry permits us to conclude that the coordinates of the other vertex of the other inscribed square are $(\frac{a^2}{a+b}.\frac{b^2}{a+b})$. I have drawn several pictures trying to see the symmetry, but I can't seem to discern it, so I left no other option but a (relatively) brute-force method. Perhaps someone with a more geometric eye could kindly point out the symmetry being appealed to.
 [/INST] We can draw a picture of this situation as follows:

We need to find $c,d,e,f$ to know the interior vertices of the squares.
Can you see that this is rotationally symmetric, if we spin it around by $180^\circ$, it will be exactly the same problem?
This means that the distance between $(0,b)$ and $(c,f)$ is equal to the distance between $(e,d)$ and $(a,0)$. Equally, the distance from $(c,f)$ to $(a,0)$ is equal to the distance from $(e,d)$ to $(0,b)$.
We can see that  $$(c,f)=\left(\frac{ab}{a+b}, \frac{ab}{a+b}\right)$$ by the definition of a square, and then we can find the distance from here to $(a,0)$:
\begin{align}D_1&=\sqrt{\left(a-\frac{ab}{a+b}\right)^2+\left(0-\frac{ab}{a+b}\right)^2}\\
&=\sqrt{\left(\frac{a^2}{a+b}\right)^2+\left(-\frac{ab}{a+b}\right)^2}\\
&=\sqrt{\frac{a^4}{(a+b)^2}+\frac{a^2b^2}{(a+b)^2}}\\
&=\frac{a\sqrt{a^2+b^2}}{a+b}\end{align}
Now we know that the distance from $(e,d)$ to $(0,b)$ is equal to this:
\begin{align}\sqrt{e^2+(d-b)^2}&=\frac{a\sqrt{a^2+b^2}}{a+b}\end{align}
We can repeat this with the other distances to get \begin{align}D_2&=\sqrt{\left(\frac {ab}{a+b}-0\right)^2+\left(\frac{ab}{a+b}-b\right)^2}\\
&=\frac{b\sqrt{a^2+b^2}}{a+b}\end{align}
and therefore \begin{align}\sqrt{(e-a)^2+d^2}&=\frac{b\sqrt{a^2+b^2}}{a+b}\end{align}
Now we have two simultaneous equations in $e,d$ to solve:
\begin{align}\sqrt{e^2+(d-b)^2}&=\frac{a\sqrt{a^2+b^2}}{a+b}\\
\sqrt{(e-a)^2+d^2}&=\frac{b\sqrt{a^2+b^2}}{a+b}\end{align}
If we solve these while remembering that $0<e<a$ and $0<d<b$ then we find that \begin{align}e&=\frac{a^2}{a+b}\\
d&=\frac{b^2}{a+b}\end{align}
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