(1) The circumscribed circle of the first question is to make a circle with the intersection of two perpendicular lines and a vertex as the radius. Then the angles of the same arc are equal, and the coordinates of point D (7 7,0) are found on the map.

(2)

Suppose as shown in the figure, ask the coordinates of point P when APB reaches the maximum value, and according to the law of materials, we can know that the tangent point at the bottom of point P is the maximum value when APB reaches the maximum value. Then we can know the coordinates of C (that is, the midpoint between A and B) (0, (m-n)/2) from the coordinates of A and B.

So OP=(m-n)/2, so OB=OP=(m-n)/2, then in RT△OBC, given BC and OC, we can find the length of OC, that is, under the root sign (mn), that is, the abscissa of point P.

If you don't understand anything, please ask questions and hope to adopt them!