EulerChain Cryptocurrency Mathematics Bounty 1

I am announcing cryptocurrency bounties for solving some of my favorite problems in mathematics. Here is the first one:

  • Let P=(v1, v2, v3, v4) be a simple polygon drawn on a plane.
  • The co-ordinates of the vertices v1, v2, v3, v4 are all rational numbers.
  • The lengths of the edges (v1, v2), (v2, v3), (v3, v4) and (v4, v1) are all integers.
  • The distance between v1 and v3 is an integer.

Conjecture 1: There exists a point x with rational coordinates inside P such that the euclidean distances between the pairs (x, v1), (x, v2), (x, v3), (x, v4) are all rational numbers.

Conjecture 2: Same as Conjecture 1 when the polygon P is convex.

Eulerchain bounties:

  • 100 Eulercoins for proving Conjecture 1. This implies Conjecture 2 is also true.
  • 50 Eulercoins for disproving only Conjecture 1.
  • 50 Eulercoins for proving only Conjecture 2.
  • 100 Eulercoins for disproving Conjecture 2. This implies Conjecture 1 is also false.
  • The bounties are valid till Dec 31 2021. If they are not resolved by Dec 31 2021, I will revisit this and update the bounties.

If you have any questions (or) solutions (or) counter-examples, leave a comment.

Happy Solving.

Prove or disprove this geometry conjecture

I am working on a geometry problem and it lead me to make the following conjecture:

Conjecture 1: Given any convex polygon (v1, v2, …. v_n) drawn on a plane with integer edge lengths, there is a point x inside the polygon such that the euclidean distances between the pairs (x, v1), (x, v2), … (x, v_n) are all rational numbers.

Theorem: Conjecture 1 is true for n = 3.

Simple Homework: Prove Conjecture 1 for an equilateral triangle.

Conjecture 1 seems like a very natural geometry problem. Is it well-known ? If you know any references, please leave a comment or email me.