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# Geometry

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Circles A, B, and C are externally tangent to each other. If AB = 14, BC = 18, and AC = 28, then find the radii of each of the circles.

Jun 27, 2021

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Picture: https://ibb.co/rMsSd3h

Let $a, b, c$ be the radii of $A, B, C$ respectively.

We have: \begin{align*} a+b&=14\\ b+c&=18\\ a+c&=16 \end{align*}

Add all equations: $2(a+b+c)=48$

Thus: $a+b+c=24$

Subtract $a+b=14, b+c=18, a+c=16$ from $a+b+c=24$ and you will get $c=10, a=6, b=8$

Thus $(a,b,c) = (6,8,10)$

Interesting how this is the length of a Pythagorean triple... probably a coincidence

Jun 27, 2021