This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
Put a round table into a (perpendicular) corner so that the table top
touches both walls and the feet are firmly on the ground. If there is
a point on the perimeter of the table, in the quarter circle between
the two points of contact, which is 10 cm from one wall and 5 cm from
the other, what's the diameter of the table?
Consider the +X axis and the +Y axis to be the corner. The table has
radius r which puts the center of the circle at (r,r) and makes the
circle tangent to both axis. The equation of the circle (table's
(x-r)^2 + (y-r)^2 = r^2 .
This leads to
r^2 - 2(x+y) + x^2 + y^2 = 0
Using x = 10, y = 5 we get the solutions 25 and 5. The former is the
radius of the table. It's diameter is 50 cm.
The latter number is the radius of a table that has a point which
satisfies the conditions but is not on the quarter circle nearest