Fair Bisectors of Triangles - Observations

We recall the old definition: A fair bisector of any convex polygon is a line cutting it into 2 pieces of equal area and perimeter.

The following are some numerical observations exclusively on triangles.

- A triangle has either 1 or 3 fair bisectors. That the number of fair bisectors should be odd is easy to show by simple continuity arguments. That it cannot be greater than 3 for triangles looks obvious although I can't quite nail a proof.

- If any triangle has 3 fair bisectors, they are concurrent - I have no theoretical justification yet for this observation.

Assuming the fair bisectors of any triangle with 3 of them to be concurrent, one can ask if this results in a new center of a triangle or if it is coincident with some already well understood center.

- And one can always ask about concurrency properties of fair bisectors of convex regions with more than 3 sides - and the max number of fbs of convex polygons with no pair of sides parallel.
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Update ( 6th May 2020): That a triangle cannot have more than 3 fair bisectors is fairly easy to show. Indeed, if a pair of edges {a,b} or a triangle support a pair of fair bisectors (that they are at most a pair follows from each fair bisector being determined by a root of a simple quadratic equation), then, it is seen that the third edge c has to be shorter than both a and b. Then, it readily follows that neither of the edge pairs {a,c} or {b,c} can support a pair of fbs. This plus the requirement that the number of fbs is odd yields the limit as 3.
Thanks to K Sheshadri for pointing out that fbs can be found by solving a quadratic equation.
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Update (9th May 2020): K Sheshadri and self did a lot of calculations and programming with a whole range of triangles. For each triangle, the fair bisectors were found to concur - with coordinate precision up to 14 decimal places in our latest attempt - and also trying to find a proof that they do. And then, yesterday, I stumbled upon this article by Shailesh Shirali: http://publications.azimpremjifoundation.org/1655/1/3_Equalizers%20Of%20A%20Triangle.pdf

The lines I have been calling 'fair bisectors' were, in fact, given the name equalizers about 10 years back and it has been well established that they do concur - at the incenter of a triangle!

Let me sign off with another observation: For convex polygons with more sides, the fair bisectors do not seem to concur. This is based only on rough estimates! Maybe one can link their intersection points to some structure such as the medial axis.