Question 59
Three circles $k_{1}$, $k_{2}$, and $k_{3}$ intersect in point $O$. Let $A$, $B$, and $C$ be the second intersection points (other than $O$) of $k_{2}$ and $k_{3}$, $k_{1}$ and $k_{3}$, and $k_{1}$ and $k_{2}$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO$, $BO$, and $CO$ intersect circles $k_{1}$, $k_{2}$, and $k_{3}$ for a second time at points $A'$, $B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that
$$
\frac{|AO|}{|AA'|} + \frac{|BO|}{|BB'|} + \frac{|CO|}{|CC'|} = 1
$$
$$
\frac{|AO|}{|AA'|} + \frac{|BO|}{|BB'|} + \frac{|CO|}{|CC'|} = 1
$$