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?

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**