Online Geometry theorems, problems, solutions, and related topics.
Proposed ProblemSee also: Complete Problem 284, Collection of Geometry ProblemsLevel: High School, SAT Prep, College geometry
The two inner circles are given by:(x-R/2)^2+y^2=R^2/4x^2+(y-R+r)^2=r^2Solve these simultaneously and set the discriminant to zero for the circles to touch each other. This gives us:-R^6+4rR^5-3r^2R^4=0 or -R^2+4rR -3r^2 =0 which can be solved to obtain r = R/3 or r=R of which only the first solution is admissible.Ajit: email@example.com
We can use Pythagorean Theorem to solve it.Since D, E, C are collinear,OD=R-r, OR=R/2 and DC=r+R/2,So solving (R-r)^2+(R/2)^2=(r+R/2)^2 will give r=R/3.