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In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of \(\triangle ABC\), as shown.
The radius of the circle with center R is decreased so that

the circle with center R remains tangent to BC,

the circle with center R remains tangent to the other three circles, and

the circle with center P becomes tangent to the other three circles.

The radii and tangencies of the other three circles stay the same. This changes the size and shape of \(\triangle ABC\). r is the new radius of the circle with center R. r is of the form \(\frac{a+\sqrt{b}}{c}\). Find \(a+b+c\).

 Dec 30, 2018

Does anyone have any  input for this one?


I have spent a lot of time looking.

If anyone else wants to step in, even with just thoughts, that would be good :)



Does anyone know how to graph a cirlce (B) centred on the circumference of another cirlce (A) and moving on that arc?

(Like a planet going around the sun)

 Jan 1, 2019
edited by Melody  Jan 1, 2019

I can give you the parametric equations for that plot.


What graphing tool are you using?

Rom  Jan 2, 2019

Thanks Rom,

I do not know how to do that but I did end up making the diagram.

I only know how to use Desmos and GeoGebra   indecision


One trouble is, I actually do not understand what the question is asking. 

Are a,b and c the side lengths of the triangle ?


Anyway, here are my graphs of the two most extreme r values.    that is r=1 and r=0.8



radius = 1




radius 0.8

 Jan 2, 2019


 Jan 2, 2019

Here's my take on this:




I guess what is wanted is that a = -1, b = 5 and c = 2.  However, this is a bit silly, as r can be expressed in an infinite number of ways simply by multiplying numerator and denominator by any positive number, p.  Then we would have a = -p, b = 5p2 and c = 2p.

 Jan 3, 2019

Thanks Alan, it always looks easy when someone else shows you how :)

It was sort of like that other circle one that you solved the other day. :) 


I suppose I mainly wanted to graph it but I know my graph is not quite right anyway.  I saw my error tonight.

Maybe I will fix it ...

Melody  Jan 3, 2019
edited by Melody  Jan 3, 2019

It always looks easy in retrospect! I looked at it in other, less productive ways first before I came up with the above.

Alan  Jan 3, 2019

Notice that r is the reciprocal of the golden ratio.

 Jan 3, 2019

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