If 
is the inradius of a circle
inscribed in a right triangle with sides 
and 
and hypotenuse 
, then
 |
(1)
|
A Sangaku problem dated 1803 from the Gumma Prefecture asks to construct the figure consisting of a circle centered at 
, a second smaller circle centered at 
tangent to the first, and an isosceles
triangle whose base 
completes the diameter of the larger circle through the smaller
. Now inscribe a third circle with center
inside the large circle, outside the
small one, and on the side of a leg of the triangle. It then follows that the line
.
To find the explicit position and size of the circle, let the circle 
have radius 1/2 and be centered at 
and let the circle 
have diameter 
. Then solving the simultaneous equations
 |
(2)
|
 |
(3)
|
for 
and 
gives
 |  |  |
(4)
|
 |  |  |
(5)
|
