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Three tangent circles

Assuming I only know the coordinates of the circle centers $A$ and $B$, and the three radii $R_A$, $R_B$ and $R_C$, how can I find the coordinates of the third circle center $C$?

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Hint

If you call $C(x,y)$ then:

$$(x-x_A)^2+(y-y_A)^2=(R_A+R_C)^2$$

$$(x-x_B)^2+(y-y_B)^2=(R_B+R_C)^2$$

Solve the system and get $C$.

You will find two solutions.

Can you see why that happens? Can you finish?

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    And be prepared to find two solutions. – 2017-01-29
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    @OscarLanzi And be prepared to analyse them and find out what that means. – 2017-01-29
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    This is where I got stuck, solving this system. Was not sure it was the right approach :) Will give this another try then! Thank you! – 2017-01-30
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    Subtracting both equations you get $y$ as a function of $x$ then back to any equation and get a quadratic equation w.r.t $x$ and find two roots. – 2017-01-30
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    That's exactly what I did, but because I need a general form it took me a few pages. I was able to find 2 solutions. Thank you guys! – 2017-01-30