Find radius of Circle

There is a circle C1 of Radius R1 and another circle C2 of radius R2 (R2 ≤ R1) such that it touches circle C1 internally

There is another circle C3 with radius R3 such that it touches the circle C1 from inside and the circle C2 from outside (it is given that R3 + R2 ≤ R1)

Now there is a circle C4 of radius R4 which will touch the circles C2 and C3 externally and the circle C1 internally, let's call it as C4. It is guaranteed that such a set of circles can be drawn.

After drawing the four circles, the figure may look something like this:

Now we have to draw a circle C5 which will touch the circles C2 and C4 externally and the circle C1 internally. Circles C5 and C3 are not the same. We have to find the radius R5 of this circle using the information given

In short Radii of C1,C2,C3,C4 are given then find the radius of Circle C5

• I am trying to draw the these circles from compass for any arbitrary value. What I think is that center of Circle C5 and Center of circle C3 lies on same straight line to common tangent of circle C2 and C4
– Mike
Commented Jul 4, 2015 at 11:22
• What is the source of this question? This question looks similar to another question about apollonian gasket which I have just answered and then the questioner deleted the question... Commented Jul 4, 2015 at 11:23
• The question is from online contest codechef.com/JULY15/problems/NTHCIR Can you help me out with radius of circle C5
– Mike
Commented Jul 4, 2015 at 11:28
• @user252363: I bet this question has to be locked, we cannot provide help for ongoing competions. Commented Jul 4, 2015 at 11:29
• @JackD'Aurizio no wonder the other questioner deleted the other question after he/she get the answer. In any event, I think a pointer to the Descrates' theorem will be fine. it help the OP to learn the material. Commented Jul 4, 2015 at 11:32

1 Answer

You have just to apply Descartes' theorem.

Assuming that $R_i$ is the radius of $C_i$ and $\kappa_i=\frac{1}{R_i}$, $\kappa_4$ and $\kappa_5$ are given by:

$$\kappa_1+\kappa_2+\kappa_3\pm 2\sqrt{\kappa_1\kappa_2+\kappa_1\kappa_3+\kappa_2\kappa_3},$$ so: $$\kappa_4+\kappa_5 = 2(\kappa_1+\kappa_2+\kappa_3).$$

• +1 before this question get locked ;-p Commented Jul 4, 2015 at 11:38