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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 \(a+b+c\).

Guest Dec 30, 2018

#1**0 **

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 :)

Also

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)

Melody Jan 1, 2019

#3**+2 **

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

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

Melody Jan 2, 2019

#5**+2 **

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 = 5p^{2} and c = 2p.

Alan Jan 3, 2019