A Formula For Prime Pythagoreans

Warning: This is a long text, so if you don't have the patience to read the whole text, don't leave rude comments
Well, let me start from the beginning, about 3 months ago I ran into a problem; The question asked by the problem was: "Display the number 1 + square root of three on the number line."
Well, I said to myself, if I want to count all the real numbers to find this number, I will look like a fool, I quickly went to geometry and said that if I wanted to find that number, I have to draw a triangle, if I draw an arc from its highest vertex with a caliper to the side of the axis of numbers so that the number is visible, but the question here was, what sides should our triangle have? We know that by drawing a bow, we add one unit to the chord, so the chord of the triangle must be the square root of three. The rest of the sides are factors of three, but if we want to put these in the Pythagorean formula, we will reach the number four, which comes from the following definition:
$$c = \sqrt{a ^ 2 + b ^ 2}$$
Now what is the problem, we found that the chord should be root three and not root four, which is equal to 2, so what should we do? Pythagoras disappointed us.
No matter what we do, nothing comes out of the presented geometrical solutions. Of course, if we draw the same triangle twice on the main triangle and make an arc, we will get the number, but the Pythagorean formula, which has been proven so much, is now unable to solve this problem. And if we want to be strict and say that there must be an algebraic solution, what should we look for? Well, I have the answer for you:
$$a ^ 2 + (b ^ 2 - \frac{a^2}{a})=\sqrt{c}$$
This is a formula that will be able to solve any problem like the above problem. This formula gives us the other sides of the triangle with which we can make a triangle with a root chord of 3.
Well, I have concluded that this formula is acceptable with all efforts and mistakes in 1 month and questions and answers from artificial intelligence, putting the formula and discussing it in the forms to my math teacher in 3 months. It is to solve any problem that is similar to our problem.
It can be solved by giving our problem to this formula, and the only condition is to assume that you do not know the second factor of the number under the square root of the chord of the triangle, and put it as b, and the first factor, which is 1 (because the number 3 is prime) put a and c as the root number in the chord which is 3.
In short, I don't want to give you a headache, you can solve the formula yourself and see that in the end you get this answer, the fourth root of the number 3.
Now, if anyone has an opinion and an idea how to use algebra, geometry and trigonometry to prove this formula, please tell me.

 

I'm checking out Pythagorean triples with a prime twist. There isn't a direct way to make them with only prime numbers, but I'm learning about the basic formulas. One formula uses two numbers, m and n, where m is bigger than n, both are positive, and they don't share common factors. The formula looks like this:

a = m^2 - n^2,
b = 2mn,
c = m^2 + n^2.

It's cool to see how these numbers work together, and it's neat that one of them has to be an even number. I'm excited to see what else I can discover as I dig into this math stuff.