(2007? Shaoxing) as shown in the figure, in the plane rectangular coordinate system xOy, O is the origin, and the coordinates of point A and point C are respectively

(1) Because of rotation, OA = BC, AB = OC, two groups of quadrangles with equal opposite sides are parallelograms, which proves that,

(2) Therefore, it is easy to find the coordinate of point B as (3,3 √ 3).

And by parabola y=ax2-2√? 3x？ =x(ax-2√？ 3) = 0，x 1=0，x2=2√？ 3/a=2(x 1 should be the abscissa of a)

So a=√? three

? So the parabola is y=√? 3x^2-2√？ 3x？ Therefore, the generation check of b coordinate satisfies that b is on a parabola.

(3) Can calculate the coordinates of point D (1,-√? 3)

tan∠OAB=-tan∠BAx=-3√？ 3=-tan∠OPD=tan∠APD

? Let P(x, 0), then tan∠OPD =∞? 3/(x- 1)=3√？ 3, so x=4/3.

(4) As shown in the figure below, because DE = 1 and AP = 1, P( 1, 0) is obtained.