As shown in figure 1, in the plane rectangular coordinate system, the vertex O of the right-angled OABC is at the coordinate origin, the vertices A and C are on the positive semi-axes of the X axis and Y axis respectively, OA=2, OC= 1, the diagonal AC and OB of the rectangle intersect at point E, and the straight line passing through point E intersects with the edges OA and BC at points G and H respectively.

(1)① Write the coordinates of point E directly: _ _ _ _ _ _ _

② verification: ag = ch.

(2) As shown in Figure 2, an arc with O as the center and OC as the radius intersects OA and D. If the straight line GH and the circle where the arc CD is located are tangent to a point F in the rectangle, find the functional relationship of the straight line GH.

(3) Under the conclusion of (2), there is a little P in the trapezoid ABHG. When ⊙ P is tangent to HG, GA and AB, find the radius of ⊙ p. 。

This topic comprehensively investigates the properties and judgments of rectangle, congruent triangles, similar triangles, tangent, linear function and Pythagorean theorem. This topic is comprehensive and difficult, but it is also a good one.

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