Let (a,r) and (r,b) be the centers of the circles tangent
to the xaxis and yaxis respectively (where r is the
radius desired). Let (c,c^{2}) be the point of tangency on
the parabola (WOLOG the first quadrant branch will
be used). The centers of the circles must lie on the
normal line at the point (c,c^{2}) (see my reply to Charlie's
post for a correction to the problem): c^{2}r c^{2}b 1
 =  =  (1)
ca cr 2c
The distance from the centers to the point (c,c^{2}) is r: (c^{2}r)^{2} + (ca)^{2} = (c^{2}b)^{2} + (cr)^{2} = r^{2} (2)
Combining (1) and (2) to eliminate the coordinates
a and b to give: (c^{2}r)^{2} + 4c^{2}(c^{2}r) = r^{2} (3)
and
(cr)^{2}/4c^{2} + (cr)^{2} = r^{2} (4)
Which in turn give r = wc^{2}/(w+1) (5)
and
r = wc/(w+2c) (6)
where w^{2} = 4c^{2}+1.
Combine (5) and (6) to find c: wc^{2}/(w+1) = wc/(w+2c)
⇒ 8c^{2}9c+2 = 0
⇒ c = [9+s√(17)]/16 (7)
where s = ±1.
The point (c,c^{2}) is the midpoint of the line segment
joining the centers of the circles. Therefore, c = (a+r)/2 (8)
Combining (1) and (8) gives c^{2}r 1
 =  (9)
cr 2c
Therefore, c(2c^{2}1)
r =  (10)
2c1
Combining (7) and (10) gives r = 3[23s√(17)]/128 (11)
QED
