We use Casey's theorem on the system formed by the three vertices of the triangle and the small circle. Let $A,B,C$ be the vertices of the triangle, and let $t_a,t_b,t_c$ be the lengths of the tangents from $A,B,C$ (respectively) to the small circle. Then, if we assume $t_a\ge t_b\ge t_c$, we get $t_a\cdot BC=t_b\cdot CA+t_c\cdot AB\ (*)$. Since $BC=CA=AB$, we can simplify $(*)$ by $BC=CA=AB$ to get $t_a=t_b+t_c$.

What is caseys theorem?

i think there is a mistake in the solution . it should be $t_a + t_c$ = $t_b$