Known: as shown in the figure, in the plane rectangular coordinate system xoy, the side length of square ABCD is 4, its vertex a moves on the positive half axis of X axis, and vertex D moves on the Y axis (points a and d do not coincide with the origin), B and C are in the first quadrant, diagonal AC and BD intersect at point P, connecting Op Q: when OA

Known: as shown in the figure, in the plane rectangular coordinate system xoy, the side length of square ABCD is 4, its vertex a moves on the positive half axis of X axis, and vertex D moves on the Y axis (points a and d do not coincide with the origin), B and C are in the first quadrant, diagonal AC and BD intersect at point P, connecting Op Q: when OA

It is proved that PM is perpendicular to x-axis and PN is perpendicular to y-axis
PM ⊥ X axis, PN ⊥ Y axis, so PM ⊥ PN, ∠ MPN = 90
P is the intersection point of the square diagonal, so ∠ DPA = ∠ MPN = 90, and AP = DP
∠MPA=∠DPA-∠DPM
∠NPD=∠MPN-∠DPM
Therefore, MPA = NPD
In △ MPa and △ NPD
∠MPA=∠NPD
∠AMP=∠DNP=90
AP=DP
So △ MPa ≌ △ MPa ≌ NPD.PM=PN
In the quadrilateral ompn, mon = PMO = PNO = 90, so it is a rectangle
And PM = PN, a group of adjacent sides of the rectangle are equal, so it is a square
OP is a square ompn diagonal, so ∠ DOP = 45
Because ∠ DOA = 90, Op bisects ∠ DOA