#Finding the radius of a circle

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spring fog
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From what you stated about h_x and p_y, I will just assume C is the origin

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and that CF is a radial segment

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So basically given the coordinates of H and P, we want the radius r, i.e. length of CF, i.e. distance of the origin from HP

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Then the equation of the circle will simply be x^2 + y^2 = r^2

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Slope of HP = (p_y - h_y)/(p_x - h_x) = -h_y/p_x

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Slope of CF = p_x/h_y

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Solving for the coordinates of F,
f_y = p_x/h_y * f_x = -h_y/p_x * f_x + h_y

sand garden
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Most of the information here is superfluous. All we need are the coordinates of f and c.

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Then we just use the distance formula.