If
is the inradius of a circle
inscribed in a right triangle with sides
and
and hypotenuse
, then
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(1)
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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
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(2)
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(3)
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for
and
gives
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(4)
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(5)
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