# Inside a sector of a big circle , there are two touching circles. Find the radius of one of them.

Consider sector of a circle $$OAB$$.
Circle with center $$M$$ touches $$OA$$ at $$P$$, $$OB$$ a $$Q$$ and arc $$AB$$ at $$N$$.
Circle with center at $$L$$ touches $$OA$$ at $$C$$, $$OB$$ at $$D$$ and circle with center $$M$$ at $$K$$.
$$OA = 98$$. $$MP = 21$$.
Find $$LC$$.

I can't even begin. I think $$O , L , M$$are colinear and then $$\triangle OCL$$ and $$\triangle OPM$$ are similar.I don't know what to do. Please give some starting hint.

From the figure, by similar triangles, (x+r) : r = (x + 2r + R) : R

Then, $x = \frac {2r^2}{(R - r)}$

Therefore, $2R + 2r + \frac {2r^2} {(R - r)} = 98$ with R = 21

Solve the corresponding equation to find r. And r should be 12.

• I got the answer as 12. Is it right? Also thanks a lot. – A Googler Apr 2 '14 at 17:22
• Yes, as shown in my edited version. – Mick Apr 2 '14 at 17:25
• Same way as mine. – Ajay Apr 3 '14 at 1:44
• Q55 Mailed to you. – Ajay Apr 3 '14 at 1:45
• @Ajay Thanks a lot! – A Googler Apr 3 '14 at 8:50