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Question: Three circles with centers respectively $A,B,C$ are mutualy tangent. Express the radius of circle with center A in terms of BC, AC, and AB respectively.

picture of tangent circles

My work: Let circles A, B, and C have radii a, b, and c, respectively and let $AB=x, AC=y, BC=z$. From here, I am not sure what to do. I tried systems of equations, but it didn't work.

Could someone please tell me what to do from here so that I can get the radius of circle A (a) and the reason behind it. I understand that there is a guess and check method to get the correct answer. However, it would be more helpful if someone could explain the rest of the algebra that I have to do to get to the answer.

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Hint

$$\begin{cases}a+b=|AB|\\ a+c=|AC| \\ b+c=|BC|\end{cases}$$

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$AC=r_a+r_c$

$AB=r_a+r_b$

$BC=r_b+r_c$

Subtract third equation from the first two.

$2r_a=AC+AB-BC$

$r_a=\frac{AC+AB-BC}{2}$

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as you have assumed that a,b,c are the radius of the circles A, B, C respectively

you can say that $$AB=a+b$$

$$AC=a+c$$

$$BC=b+c$$

thus $ BC-b=c$

put in the second equation

$AC=a+(BC-b)$

$AC-BC-a=-b$

$BC-AC+a=b$

put in equation 1

you get $AB=a+(BC-AC+a)$

$$2a=AB-BC+AC$$

$$a=\frac{AB-BC+AC}{2}$$

as equation 1 says $AB=a+b$

thus put a and get the value of b

similarly, solve for b and c

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  • $\begingroup$ Hi James, thanks for helping me. From here, what should I do to find a, which is the radius of circle A? $\endgroup$ – T.Mok Aug 15 '18 at 14:19
  • $\begingroup$ see the edited answer three variables and three equations $\endgroup$ – Deepesh Meena Aug 15 '18 at 14:24

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