Find radius of circle inscribed between y axis, unit circle, and y=sqrt(x).

nerd_snipe.py

# Solution to https://twitter.com/iwontoffendyou/status/1704935240907518367
# Tobin Fricke 2023-09-22

import numpy as np
import scipy.optimize

def dist_from_circle_center_to_curve(R):
    # Center of circle of radius R that is tangent to the y axis and the unit circle.
    x0 = R
    
    # The point of tangency of the small circle with the big unit circle is found by drawing
    # a line through their centers. From this we see that the center of the small circle is
    # a distance (1-R) from the center of the unit circle. We can now draw a right triangle
    # and use Pythagoras to get the y-coordinate of the origin of the small circle:
    y0 = (1 - 2*R)**0.5
    
    # What is the shortest distance from the point (x0, y0) to the curve y = sqrt(x)?
    
    def dist_from_circle_center_to_curve(x):                
        # The curve y = sqrt(x)
        y = x**0.5
        
        # Distance from the circle center to the curve.
        return ((x0 - x)**2 + (y0 - y)**2)**0.5
    
    # Find the point on the curve closest to the origin of the circle.
    _, fopt, _, _, _ = scipy.optimize.fmin(func=dist_from_circle_center_to_curve,
                                           x0=0.5, full_output=True, disp=False)
    return fopt

scipy.optimize.fixed_point(dist_from_circle_center_to_curve, x0=0.25)