If δ BCG δ ach, then δ∠BCF =∠ACH.

∠ ECF = 30，∴∠ BCF = 30。

(2) in ∵ RT △ ABC, AC = AB = 2, ∴ AB = 2 √ 2, and the height on the side of AB (set to H) H = √ 2.

∵GH=x,∴BG+HA=2√2-x

Area of y = △ gch = 1/2? x？ h=√2x/2

That is, the functional relationship between y and x is: y = √ 2x/2.

When x is the largest, y is the largest.

When CG⊥AB, CG is the smallest and X is the smallest; When CH and CA coincide, CH is the largest and x is the largest.

At this time ∠ CGH =180-(30+45) =105, and x/sin30 = 2/sin 105 is obtained by sine theorem.

x = 2/sin 105×sin 30 = 2/sin 75×sin 30

=2/[(√6+√2)/4]×√3/2

=3√2-√6

That is, when x = 3 √ 2-6, y is the largest.

Y max = √ 2/2× (3 √ 2-√ 6) = 3-√ 3。

Note: SIN 75 = SIN (45+30) = SIN 45 COS30+COS 45 SIN30.

=√2/2×√3/2+√2/2× 1/2

=(√6+√2)/4