In a triangle if $P\left(\frac{1}{a-b+x}+\frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)=\frac{1}{2}$ then $x=\pm\frac{3}{4}, \pm\frac{3}{5}$ or $0$?

Question

Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let

$$ f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right) $$

The plot of $x$ vs $f(x)$ is show below along with line $\displaystyle y = \frac{1}{2}$.

If we zoom in further, we see that the curve $y = f(x)$ intersects the half line $\displaystyle y = \frac{1}{2}$ at five points as shown below.

Experimental data show that the all the intersections occur at rational points.

Question: Is it true that $\displaystyle P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right) = \frac{1}{2}$ at $\displaystyle x = \pm\frac{3}{4}, \pm\frac{3}{5}$ and $0$?

Related question: Does every triangle satisfy $\frac{1}{a-b+\pi R} + \frac{1}{b-c+\pi R} - \frac{1}{c-a+\pi R} < \frac{1}{R}$?

Julia code to estimate $f(x)$:

step = 10^7
x = -4.83

while x <= 4.83
    count = 0
    f = 0
    
    while count < step
        count += 1
        
        # Generate random angles for the vertices
        angles = (rand(3) .* 2 * π)
    
        # Calculate the coordinates of the vertices on the unit circle
        vertices_x = cos.(angles)
        vertices_y = sin.(angles)
    
        # Close the triangle by repeating the first vertex
        push!(vertices_x, vertices_x[1])
        push!(vertices_y, vertices_y[1])
    
        # Calculate the side lengths using vectorized operations
        x_diff = diff(vertices_x)
        y_diff = diff(vertices_y)
        side_lengths = sqrt.(x_diff.^2 + y_diff.^2)
    
        a = side_lengths[1]
        c = side_lengths[2]
        b = side_lengths[3]
        
        if 1/(x+a-b) + 1/(x+b-c) > 1/(x+c-a)
            f += 1
        end
    end
    
    println(x, " ", f," ", string(f/count)[1:end-1], count," ")
    
    x = round(x + 0.01, digits=3)
end