In the smallest of the three triangles (with sides [tex]a,b,3[/tex]) we have
[tex]a^2+3^2=b^2[/tex]
In the second-largest triangle (with sides [tex]a,108[/tex]), the length of the missing side is [tex]\sqrt{a^2+108^2}[/tex]. Then in the largest triangle (with sides [tex]b,111,\sqrt{a^2+108^2}[/tex]), we have
[tex]b^2+\left(\sqrt{a^2+108^2}\right)^2=(108+3)^2=111^2[/tex]
[tex]\implies b^2+a^2+108^2=111^2[/tex]
[tex]\implies b^2+a^2=657[/tex]
Equivalently, we have
[tex](a^2+9)+a^2=657\implies 2a^2=648\implies a^2=324\implies a=18[/tex]
In turn, we find
[tex]b^2=18^2+3^2=298\implies b=\sqrt{333}=3\sqrt{37}[/tex]
