Friends:
Suppose that $\Gamma_{1}$ and $\Gamma_{2}$ are two circumferences that are externally tangent (they have exactly one point in common and neither of them is contained in the region bounded by the other one). Let us denote with $P$ the point of contact of these two circumferences. Prove that the tangent to $\Gamma_{1}$ that passes through $P$ coincides with the tangent to $\Gamma_{2}$ that passes through $P$.
How would a proof of this assertion from first principles go?
Thanks in advance for your suggestions.