+25 Externally tangent circles with centers at points $A$ and $B$ have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$, where $B$ is on $\overline{AC}$. What is the length $BC$?
+128485 Let the tangent line touch circle A at D and circle B at E
Let the distance from B to C = x
We can set up similar triangles ADC and BEC such that
AD / AC = BE / BC
5 / ( 5 + 3 + x) = 3 / x
5 ( 8 + x) = 3 / x cross-multiply
5x = 3 (8 + x)
5x = 24 + 3x
2x = 24
x = 12 = BC
