The circle touches both coordinate axes.
2. You are dealing with a right triangle where the circumcentre is the midpoint of the hypotenuse.
3. If the tangentpoint is T(t, p) then the tangent has the equation:
4. If the tangent is parallel to the coordinate axes then there doesn't exist a circumcentre. Therefore the straight lines x = a and y = a must be asymptotes of the locus.
5. I've drawn a sketch: Triangles in blue, locus in red, asymptotes in green