# 3 circles with radii 66, 77, and 88 externally tangent to each other, find the radius of the circle internally tangent to the other circles.

In the diagram, we see that there are 3 circles that are all externally tangent to each other and internally tangent to a much bigger circle. The radii of the 3 smaller circles are 66, 77, and 88. What is the radius of the bigger circle?

I actually solved this problem by drawing in the center of the bigger circle, drawing lines from that center to the 3 points of tangency on the large circle, then set up a system of equations and go for there (if you want me to be more specific just ask) but this is really messy. Is there a really nice intuitive way to go about this, or am I stuck doing the messy work?

$$(\frac 1{r_1}+\frac 1 {r_2}+\frac 1{r_3}+\frac 1 {r})^2=2(\frac 1{r_1^2}+\frac 1 {r_2^2}+ \frac 1{r_3^2}+\frac 1{r^2})$$
Where $$r_1, r_2, r_3$$ are radii of given circles and r is the radii of small(in the middle of circles) and big circle(the one you try to find it's radius).. That is you get a quadratic equation in term of r , solving it gives you the radius.