There are similar questions and answers out there but other answers I have seen seem unnecessarily complicated. I came up with what I think is a simple approach some time ago but I'm now doubting my past reasoning.
Three spheres have centers at the vertices of a Pythagorean triple and they touch at points along the sides of that triangle. How do we find the radii of these spheres?
For the diagram below, is this logic sound? I cannot remember how I arrived at the conclusion in equation (1). Can anyone help reconstruct the logic?
\begin{align*}
R_1+R_2=8\qquad
R_1+R_3&=15\qquad
R_2+R_3=17\\
\\
\implies (R_1+R_3)-(R_1+R_2) &=(15-8)=7\\
&=(R_3-R_2)\tag{1}\\
\implies (R_2+R_3)+(R_3-R_2) &=2R_3= (17+7)=24\tag{2}\\
\implies R_3=\dfrac{24}{2}&=12\tag{3}\\
\implies R_2=(17-12)&=5\tag{4}\\
\implies R_1=(8-5)&=3\tag{5}
\end{align*}
