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Largest circle in quadrilateral (Posted on 2021-09-17) Difficulty: 4 of 5
Given a quadrilateral with side lengths 2, 4, 5, and 6 units, what is the radius of the largest circle which can be drawn completely inside of it?

No Solution Yet Submitted by Danish Ahmed Khan    
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soln(s) | Comment 5 of 8 |
I did write the program just I outlined in my first post 
"Possible Approach". Yes, each of the three orderings 
gives its own solution: there was a largest incircle 
for each of the three cases. And, yes, my triangle 
"shortcut" approach was too good to be true.   
For that case, with ordering of sides: 2,4,5,6, the largest 
incircle is larger than 1.604. It is about 1.7639. 
And yes, the best of the three orderings is a nearly 
cyclic quadrilateral with ordering 2,5,6,4 (as per Brian) 
and has an incircle radius shown below. It's interesting 
that the quadrilaterals with the largest incircle are not 
exactly cyclic. I checked this using double precision.       


Four Sides: 2.  4.  5.  6.  
Four Points: (0.0000,0.0000)(-2.5163,3.1094)(1.5741,5.9849)(2.0000,0.0000)  
Four Angles 1-2,2-3, etc: 128.982   86.124   58.964   85.929  
Incenter (3.8390595213059098E-002,2.7523426988620265)  
Incircle radius: 1.76128707  
Opposite angles added: 187.946  172.054      

Four Sides: 4.  2.  5.  6.  
Four Points:  (0.0000,0.0000)(-1.1028,1.6685)(1.9407,5.6355)(4.0000,0.0000)  
Four Angles: 123.463  109.042   57.568   69.927  
Incenter  (1.3600505196827819,1.8459933539353104)  
Incircle radius: 1.84599335  
Opposite angles added: 181.031  178.969    

Four Sides: 4.  5.  2.  6.  
Four Points:  0.0000  0.0000   0.3474  4.9879  2.2064  5.7256   4.0000  0.0000  
Four Angles: 86.016  115.630   85.748   72.606  
Incenter (1.7623691639765311,1.6438902087165359)  
Incircle radius: 1.64389021  
Opposite angles added: 171.764  188.236      

Edited on September 24, 2021, 3:10 pm
  Posted by Steven Lord on 2021-09-22 06:34:48

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