Selection and Training Session

Let $R$ be the radius of the given circle, $M$ the midpoint of $AC$, $N$ the midpoint of $BM$, $T$ the midpoint of $BK$. Let $a,b,c,m,n,t,k$, $x$ be the (complex) coordinates of the points $A,B,C,M,N,T,X$, respectively. Then $m=(a+c)/2$, $n=(b+m)/2$, $k=(c+x)/2$, $t=(k+b)/2$. So we can easily find $x=4t-2b-c$, $n=(2b+c+a)/2$. The complex equation of the given circle is $|x-a|=R$, i.e. $|4t-2b-c-a|=R$, or $|t-(2b+c+a)/4|=R/4$, or $|t-n|=R/4$. That is the locus of points $T$ is the circle of radius $R/4$ with the center at $N$.