Product of slopes of rational points on the unit circle (related to pythagorean triples)

Question

The slope of the line from the origin through a rational point on the unit circle can be written as

$$s = \frac{2t}{1-t^2}$$ where t is a rational parameter and $-1<t<1$

I conjecture that for slopes greater than 0, the product of any two of these slopes is unique, meaning if $s_1s_2=s_3s_4$, then $s_1=s_3$ and $s_2=s_4$. I believe this amounts to showing that:

$$\frac{4t_1t_2}{(1-t_1^2)(1-t_2^2)} = D$$ where D is any rational positive constant, has only 1 solution pair $(t_1,t_2)$ for rationals $t_1,t_2>0$.

Are there techniques to tackle this problem? Is this conjecture false? Appreciate any tips.

Answer 1

I think that the conjecture is false.

For $D=1$, take $$(t_1,t_2)=\left(\frac{1-s}{s+1},s\right)$$ for $s\in\mathbb Q$ such that $0\lt s\lt 1$.