Right-angled triangles have $3$ excircles. I'm struggling to find a formula which gives me the radius of all three excircles. I've been struggling with this for a while. I've done some Googling, and I think I have parts of the correct formula:
\begin{align} s &= \frac{a+b+c}{2} & A &= \sqrt{s(s-a)(s-b)(s-c)} \\[0.5em] r_1 &= \frac{A}{s-a} & r_2 &= \frac{A}{s-b} \hspace{3em} r_3 = \frac{A}{s-c} \end{align}
$A$ is the area of the right-angled triangle; $a$, $b$, and $c$ are sides of the right-angled triangle; $s$ is the semi-perimeter of the right-angled triangle; $r_1, r_2$ and $r_3$ are the radius of the excircles.
Can anyone find the formula? (Preferably I would like a formula without using any angles.)
Not a problem:), and this is how you can arrive at the result!
We know that $\Delta = \frac{abc}{4R}$
Now I will assume that $c$ is the hypotenuse. Draw a circum-circle around your triangle you can easily observe by Thales theorem that $c$ is the diameter of the circle . Hence $c=2R$, Therefore $\Delta = \frac{ab}{2}$ or Area is triangle = $\frac{1}{2}$ * base * height.