The circle passing through three distinct points $(1,t),(t,1)$and $(t,t)$ passes through the point 

: Equation of a circle passing through $(1, t)$ and$(t, 1)$ is

$(x-1)(x-t)+(y-t)(y-1)+\lambda \left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}y\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}t\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ t\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right|=0$

It passes through$(t, t)$  if $0+0+\lambda \left|\begin{array}{ccc}t& t& 1\\ 1& t& 1\\ t& 1& 1\end{array}\right|=0$

$\Rightarrow \lambda =0$  

 The circle through the given points is  $(x-1)(x-t)$

$+(y-t)(y-1)=0$ , which clearly passes through $(1,1)$