Two circles with equal radii are externally tangent
at a point on the parabola y = x^{2}. One of the circles
is also tangent to the x-axis while the other is also
tangent to the y-axis. Find the radius of both circles.

One root seems to agree with Charlies numerical results, giving:

r = 3(23 - sqrt 17)/128 ~ 0.442427 T = ((9 + sqrt 17)/16, (49 + 9 sqrt 17)/128) ~ (0.820194, 0.672718) Upper circle: centre at (0.442427, 0.90301) and touching the y axis Lower circle: centre at (1.19796, 0.442427) and touching the x axis

and the other root also gives a valid solution:

r = 3(23 + sqrt 17)/128 ~ 0.635698 T = ((9 sqrt 17)/16, (49 - 9 sqrt 17)/128) ~ (0.304806, 0.0929066) Upper circle: centre at (-0.026086, 0.635698) and touching the x axis Lower circle: centre at (0.635698, -0.449885) and touching the y axis