[Guest]

From the intimate circles riddle we have concluded that the maximum number of circles on a 2D plane that you can draw with each circle being tangent to every other circle is 4 and you have two configurations for it, 3 circles forming a triangle and the 4th will go either inside it or outside.

Now you draw 3 circles that are tangent to each other, their radiuses are R₁ R₂ and R₃, to make things simple R₁≥R₂≥R₃...

Easy:

What does R₃ has to be so you'd be able to put in a 4th circle as both combination? (in terms of R₁ R₂)

Not Easy:

What is the 4th circle's radius (R₄) in each combination? (in terms of R₁ R₂ R₃)

[Guest]

since any 3 points can define a circle, the relationship between the three radiuses r1 r2 and r3 can be practically anything. if you have a solution to this i would very much like to see it.

[EventHorizon]

The only value for R₃ that I see causing problems is such that R₄ needs to be a line on the outside case. For all other cases a circle can be put on the inside or outside.

There are cases where the outside circle is wholly on one side instead of around the others, but there's always two circles that can be placed (except the line case which would require infinite radius).

I'll work on "not easy" later.

Edited October 6, 2010 by EventHorizon

[fabpig]

Edit....Sorry messed up. I'll try again in a sec

Edited October 7, 2010 by fabpig

[EventHorizon]

The math got kinda nasty, but it seems like I may have the answer.

I calculated it using the assumption that R₁ + R₂ = 1, so scale down then scale back up.

-2R₁R₂R₃²-2R₁²R₂²R₃ +/- 4R₁R₂R₃ * sqrt(R₁R₂R₃(1+R₃))

--------------------------------------------------------------------------------------

4R₁R₂R₃ + 8R₁R₂R₃² - 2R₃² - 2R₁²R₂²

It worked for the two situations I checked (both inner circle and outer circle by using the +/-, and using negative radius values as positive). I was actually surprised it worked for those cases because it didn't look quite right to me (and because it took 3 pages of hand written algebra that I could have easily introduced errors into).

I just tried a new situation... it got the inner circle, but didn't get the outer.

So, can anyone verify/disprove this equation? When does it get the outer circle's radius right?

After a little searching online I found a better solution...

http://en.wikipedia.org/wiki/Descartes%27_theorem

[Guest]

The only value for R₃ that I see causing problems is such that R₄ needs to be a line on the outside case. For all other cases a circle can be put on the inside or outside.

There are cases where the outside circle is wholly on one side instead of around the others, but there's always two circles that can be placed (except the line case which would require infinite radius).

I'll work on "not easy" later.

Oh, sorry looks like I missed a big point here, I was thinking of the case where the smallest circle is smaller than the case you mentioned that you can't put a big circle that'll be tangent with all the others from the inside, I didn't think about the option of putting it from the outside...

[bonanova]

Easy:

What does R3 has to be so you'd be able to put in a 4th circle as both combination? (in terms of R1 R2)

Not Easy:

What is the 4th circle's radius (R4) in each combination? (in terms of R1 R2 R3)

R₃ can be anything, if I understand the question: [that there is both a larger and a smaller fourth circle.]

From the where R₁=R₂=1 and R₃=1/4, the larger fourth circle is a line.

Given R₁=R₂=1, that value of R₃ divides the two cases:

If R₃>1/4, the fourth circle will enclose the first three.

If R₃<1/4. the first three will be external to the fourth.

The smaller fourth circle lies in the region exterior to, but enclosed by, the first three in either case.

The general formula relating the four radii that I asked about is Descartes Theorem, found about by Event Horizon.

Edited October 8, 2010 by rookie1ja
font edited as per request