Problem

Source: Federation of Bosnia and Heryegovina, 3rd grades, 2008.

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Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$ at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$. Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$.