Three circles of radius 6, 7, and 8 are externally tangent to each other. There exists a smaller circle tangent to all three (in the space created between the three original circles).
What is the radius of this smallest circle?
Make a triangle with the centres of each of the 3 large circles. This triangle has sides of dimension 13, 14 & 15 units.
The small centre circle has radius R.
Divide the large triangle into 3 smaller triangles, with the common corner being the centre of the small circle. This makes the sides of the smaller triangles:
T1 : 13, 7+R, 6+R
T2 : 14, 8+R, 6+R
T3 : 15, 8+R, 7+R
Using area = sqrt[s(s-a)(s-b)(s-c)],
where a,b,c are sides of triangle and s=(a+b+c)/2
Area large triangle = sqrt(22.214.171.124) = 84
Area T1 = sqrt[(13+R).R.6.7] = sqrt(42R^2 + 546R)
=> Area T1 ~ sqrt(546R) for small R
& Area T2 = sqrt[(14+R).R.6.8] ~ sqrt(672R)
& Area T3 = sqrt[(15+R).R.7.8] ~ sqrt(840R)
Sum of areas of 3 smaller Ts = area of large T
=> sqrt(546R) + sqrt(672R) + sqrt(840R) ~ 84
=> sqrt(R) ~ 84/[sqrt(546)+sqrt(672)+sqrt(840)]
=> R ~ 1.15
Edited on August 1, 2004, 7:37 pm
Edited on August 1, 2004, 7:39 pm
Posted by David
on 2004-08-01 19:29:33