An Equation That Saved the World // Një ekuacion që shpëtoi botën

                Well, not exactly the World, but at least the Problem from yesterday's Post. I promised an analytical solution there, which here it is.

          We choose a Cartesian Frame with the origin at point $O$, the $x$-axis along the side $OA$ and the $y$-axis along the side $OB$. The coordinates are marked on the figure.

          We will calculate the slope of line $AF$, showing that it is $-1$ so $\phi=135^{\circ}$.

          The equation of the line $OE$, which passes through points $O(0,0)$ and $E(a-c,b)$, is

$$y=\frac{b}{a-c} \cdot x.\tag{OE}$$

The equation of the line $CD$, which passes through points $C(0,c)$ and $D(a,b)$ is

$$y-c=\frac{b-c}{a-0} \cdot (x-0). \tag{CD}$$

Point $F$ - the intersection of lines $OE$ and $CD$ - has the coordinates obtained from solving the two equations $(OE),\;(CD):$

$$\frac{b}{a-c} \cdot x-c=\frac{b-c}{a} \cdot x\;\;\Rightarrow\;x=x_F=\frac{a(a-c)}{a+b-c};\tag{$x_F$}$$

and $y_F=y\underset{(OE)}{=}\frac{b}{a-c} \cdot x_F=\frac{b}{a-c}\cdot \frac{a(a-c)}{a+b-c}=\frac{ab}{a+b-c}. \tag{$y_F$}$

      The slope of line $AF$ is

$m=\tan \phi =\frac{y_F-y_A}{x_F-x_A} \overset{(x_F)\;(y_F)}{===} \frac{\frac{ab}{a+b-c}-0}{\frac{a(a-c)}{a+b-c}-a}=\frac{\frac{ab}{a+b-c}}{\frac{a^2-ac-a^2-ab+ac}{a+b-c}}=\frac{ab}{-ab}=-1.$

$Q.E.D.$