You may also like

problem icon

Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

problem icon

Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

problem icon

Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

Circles in Circles

Stage: 5 Challenge Level: Challenge Level:1

This solution together with the diagram was sent in by Derek Wan, age 17 of Sha Tin College, Hong Kong

Solution to Circles in Circles

To find $OA=r_1$, $OB=r_2$ and $OD=r_3$ we must examine the diagram and make appropriate considerations. By dropping a perpendicular from $O$ to $V$, the midpoint of $UC$, as the radius is perpendicular to the tangent, we can find $OA = r_1$. Since triangle $TCU$ is equilateral $\angle TCU=60^o$ and $OC$ bisects $\angle TCU$ so $\angle OCV = 30^o$.

Since $OC=OA+AC$ $${CV\over OA+AC}={\sqrt 3 \over 2}$$ As $AC=CV=1$ and $OA=r_1$, $${1\over r_1+1} = {\sqrt 3 \over 2}$$ $$r_1+1 = {2\over \sqrt 3}$$ $$r_1 = {2\over \sqrt 3}-1$$ With $r_1$ we can find $r_2$ and $r_3$. Since $OV=r_2$, $CV=1$ and $\angle OCV=30^0$, $$r_2={1\over \sqrt 3}.$$ Since $OD=OA+AC+CD$, $$r_3={2\over \sqrt 3} - 1 + 1 + 1 = {2\over \sqrt 3} + 1.$$ Now $$r_1r_2 = \left({2\over \sqrt 3 }- 1\right )\left({2\over \sqrt 3} + 1\right )$$ $$\quad = {4\over 3} + {2\over \sqrt 3} - {2\over \sqrt 3} - 1$$ $$\quad = {1\over 3}$$ $$\quad = \left({1\over \sqrt 3}\right)^2$$ $$\quad = r_2^2$$