Krdytkyu

There is one quarter circle have radius same as side of square. There is semi-circle that has diameter equal to side of square. Then find radius of circle which is tangent to these circle and side of square.

Hint: For each of the two large circles, construct the right triangle with hypotenuse connecting the center to the center of the small circle and legs parallel to the sides of the square.

Let $x$ denote the distance between the lower left corner of the square and the point of tangency between the bottom side and the small circle.

Pythagorus gives us two equations:

$$(4-r)^2+x^2=(4+r)^2quad & quad (2-r)^2+(4-x)^2=(2+r)^2$$

A geometric construction based on the use of conic sections:

• Locus of the centers of circles touching two given circles is a hyperbola, whose foci are the centers of the given circles as explained here.


• Locus of the centers of circles touching a given circle and a straight line is a parabola. Its focus is located in the center of the given circle, and the directrix is parallel to the given line, the given line being equidistant to the focus and directrix.
  
  In the enclosed figure is the focus $A$ and the straight line $CD.$ The directrix lies bellow, and is out of the picture.

GeoGebra gives a nice picture, but the computation of the radius based on this method would be rather boring.

Hint: My attempt would be to formulate the equations for the three circles and the line segement and then check if there is a common solution possible. (Two intersection points with the circles, one with the line)

Circle: $(x - x_0)^2 + (y - y_0)^2 = r^2$

Line: $A x + B y = C$

Looks close to Descartes theorem, by the way.

Update:

Choosing the lower left corner as origin we get the equations:

Large circle
$$
x^2 + (y - 4)^2 = 4^2 quadquad (L)
$$

Medium sized circle
$$
(x-4)^2 + (y-2)^2 = 2^2 quadquad (M)
$$

Small circle
$$
(x-x_0)^2 + (y-y_0)^2 = r^2 quadquad (S)
$$

Line segment
$$
x in [0, 4] quad y = 0 quadquad (LS)
$$

The small circle touches the line segment from above, if the center is
$$
(x_0, y_0) = (x_0, r)
$$
for some $x_0$.