Introduction
Above are five triangles that all have the same value r and thus B, E, H, K, and N are all points on a circle with radius r. Triangle ABC is a right angled isosceles triangle.
All triangles are different and ordered descending such that: $$a>b>c>d>e>0$$
In this case we are only interested in lattice points. a, b, c, d, and e are all odd numbers. a in specific is of form (4k+1), because adding gaussian primes factors (4k+3) will only multiply every other value by the same amount.
$$r= \sqrt{2}\cdot a$$ $$a= (4k+1)$$ $$a,b,c,d,e,k\in\mathbb Z$$
Problem
Given the above information, prove that any (or a combination) of the following is impossible:
- $a^2+d^2=b^2+c^2$
- $c^2+e^2=2d^2$
- $a^2+e^2=b^2+d^2$
It can be proven by showing that:
- This cannot be done with integers;
- This can only be done with irrational numbers;
- Some trigonometry or number theory fact proving this impossible.
Solving attempts so far
I have tried numerous ways of calculus and trigonometry of solving this, but I always seem to be one piece of information or fact short. Furthermore I have set up some coding to check is solution for the main problem can be found. At time of writing the program has checked values up to r = 5.2*10^13.
I hope this problem proves plenty of challenge for you. May you be able to solve I will be greatly thankful!