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Given three circles which share the same tangent and touch each other as shown in the figure,  If the radius of the larger Orange Circle is 9 units and that of the smaller Green circle is 4 units.  What is the radius of the white circle which lies in between both the circles and also shares the same tangent ?

May 5, 2020

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We can form a 5-12-13  right triangle  by connecting the centers  of the  circles  and  then drawing a parallel line to the base of the figure  intersecting the two  vertical  lines in the figure....the length of this line = 12

And we  can  create  two more  right triangles

One on the left with  legs of  (9-r) , x   and a hypotenuse of (9 + r)

And one  ont the right with legs of (4 - r) (12-x)  and (4 + r)

And we have  this system of equations

(4 - r)^2 + (12-x)^2  = (4 + r)^2

(9 - r)^2  + (x)^2  = (9 + r)^2

These are a little sticky to solve  but using WolframAlpha  we  find the  solutions for   x  and r  to be

x  = 36/5  and  r   = 36/25  = radius of the smaller circle

So....the  equation of the  small circle  is   ( x - 36/5)^2  + ( y -36/25)  = (36/25)^2

Here's a pic : https://www.desmos.com/calculator/5yzqb20sxi

May 5, 2020