In rectangular ABCD, ab = 4, ad = 2, point m is the midpoint of AD. point E is a moving point on edge ab. connect EM and extend the intersection line CD at point F, cross m to make the vertical line of EF, the extension line of BC at point G, connect eg, and the intersection side DC at point Q. let the length of AE be x and the area of triangle EMG be y (1) Find the tangent value of ∠ MEG; (2) Find the analytic expression of Y with respect to x, and write out the value range of X; (3) The midpoint of Mg is denoted as point P and connected with CP. if {PGC} efq, find the value of Y

In rectangular ABCD, ab = 4, ad = 2, point m is the midpoint of AD. point E is a moving point on edge ab. connect EM and extend the intersection line CD at point F, cross m to make the vertical line of EF, the extension line of BC at point G, connect eg, and the intersection side DC at point Q. let the length of AE be x and the area of triangle EMG be y (1) Find the tangent value of ∠ MEG; (2) Find the analytic expression of Y with respect to x, and write out the value range of X; (3) The midpoint of Mg is denoted as point P and connected with CP. if {PGC} efq, find the value of Y

Extending GB intersection me to h, triangle MAE is similar to EBH, triangle FCH is similar to HMG, using similarity theorem, triangle side length ratio is the same, then MH = 4 * (root (1 + x square)) / x, Mg = (MH / HC) * FC = 4 * (root (1 + x square)), tangent value of MEG = mg / MH = 4; y = mg * me / 2 = 4 * (1 + x square)), x = 0 ~ 4; because CG, PG and Fe can be calculated, then FQ can be calculated, and the value of X can be calculated by proportion formula, Then I can calculate the value of Y. I'm going to do something else. I want to be specific, OK?