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?
Best Answer
Hint:
Apply Descartes theorem:
$$(\frac 1{r_1}+\frac 1 {r_2}+\frac 1{r_3}+\frac 1 {r})^2=2(\frac 1{r_1^2}+\frac 1 {r_2^2}+ \frac 1{r_3^2}+\frac 1{r^2})$$
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.