Question 199
Consider circles $(O_1)$, $(O_2)$, $(O_3)$ tangent to $d$ at $A$, $B$, $C$ and $(O_2)$ is the biggest circle, externally tangent to $(O_1)$, $(O_3)$. Let $BD$ be the diameter of $(O_2)$. The external tangent line (differs from $d$) of $(O_1)$, $(O_3)$ cuts $(O_2)$ at $X$, $Y$. Let $K$ be the midpoint of the arc $XBY$ of $(O_2)$. Prove that the circle of diameter $AC$ touches $DX$, $DY$.