Given a right triangle with sides $ABC$ make two more right triangles using sides $A$ and $C$ (long side) and a new long side $x$ (same for both new triangles). By Pythagoras the implied third sides will have lengths $a$ and $c$ such that $a^2+A^2 = x^2 = c^2+C^2$.
Now using some algebra I can show that if we can form a triangle with sides $aBc$ it must be right, too, viz.: $B^2+c^2 = B^2 + x^2 - C^2 = x^2 - A^2 = a^2$
But that feels just wrong like instrument flying on a bright day.
- either rearrange the figure in such a way as to make it ($aBc$ is right) obvious
- or make a direct geometric argument
- or a combination of both?
Note on the figure. By unfortunate coincidence (pun intended) the purple circle appears to pass through $\angle AB$. That is not necessarily the case. The circle is the one of radius $c$ around $\angle BC$