Put a round table into a (perpendicular) corner so that the table top touches both walls and the feet are firmly on the ground.

geometry/table.in.corner

Problem

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?

Show solution

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
perimeter) is

    (x-r)^2 + (y-r)^2 = r^2 .

This leads to 

     r^2 - 2r(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.  Its 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
the corner.