FREE GEOMETRY PROBLEMS ONLINE --- Please try to see the good side :)

Tony García has sent this one.  The solution is not confirmed.  The relationship may be simplified.  
At least, a more concise expression may be found - with your help...

Circles are congruent. This will be corrected later.

Roland Sampy submited this one.  Let's confirm his answer.
Happy new year to all, by the way!
 
 
 
 
by Roland Sampy
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Tony García‏ from Dominican Republic submitted this one.
It looks like it requires brute force.
Better than being idle I guess!  So enjoy...

The answer is confirmed.



OK, OK, I see world's changing.
Inaction's fun up to a certain point.
Then it becomes straight annoying.





Let's start with this simple one for high-schoolers :)
Pure-geometric solution will be appreciated.
We'll build upon this later.





Based on the evidence presented by Bleaug, this is updated as follows: 



This is just one step forward to the general case. Should not be very hard unless the claim is wrong :)
Yes, a geometric solution will be appreciated...

We'll return back to Sampy's question later. But, for a change let's see a simple one - or at least it looks simple. We're looking for the shortest proof of the claim below. May be purely-geometric one? Or is the claim wrong? Let us know...

Below is the general case submitted by Roland Sampy.
There seems to be at least 2 expressions for
r.
We had a solution, but it looked a little clumsy.  Moreover, we lost it :)
So, do help us to recover...  Oh yes, do enjoy as well...


Wooow... This has been quite long time, huh? Life makes people busy (although that is not the exact excuse.)
Is everybody alive? What have you been up to?  Bleaug?  Newzad?  Polar Fox ?? ...
 
The following fox is a simplified version submitted by Roland Sampy.
We should be able to ask a general version later.
You mean next year???
Who knows hopefully earlier. Here you are...
Oh, by the way the answer was not confirmed yet. So, the claim could be wrong.


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Happy new year to everybody!

A new year,
a new beginning,
An unheard adventure - even it makes you late for dinner,
a new beginning.

Cold air that you breath in,
A warm shoulder that you can lean
A promise you've given to yourself - one more time,
a new beginning.

A morning sky painted in the East,
A colorful sunset as it fills the West,
Bread, cheese, water - and the rest,
a new beginning.

 
 
 
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 Some old foxie...

 
 
 
 
 
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(Actually we haven't been updating this for looong time :)

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1 Chicken and 8 Foxes  - by - Karen Strohbeen

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