7. In the plane of square $A B C D$, there is a point $P$ such that $\triangle P A B, \triangle P B C, \triangle P C D, \triangle P D A$ are all isosceles triangles. Then, the number of points $P$ with this property is
A. 9
B. 17
C. 1
D. 5
Answer
As shown in the figure, point $P$ must lie on the perpendicular bisectors of the four sides of the square, except for point $O$. Considering the case where the side length of the square is the waist length. Draw circles with $A, B, C, D$ as centers and the side length as the radius, obtaining 8 intersection points.

Thus, there are 9 points $P$ with such properties, so the answer is $A$.