∴∠BCD=45 =∠ABC，∠A+∠DCA=90，∠A+∠ABE=90，

∴DB=DC,∠ABE=∠DCA，

∫In△DBH and △DCA

∠BDH=∠CDA BD=CD ∠HBD=∠ACD，

∴△DBH≌△DCA，

∴BH=AC.

(2) connecting CG,

F is the midpoint of BC, DB=DC,

∴DF vertically divides BC,

∴BG=CG，

∵∠ABE=∠CBE,BE⊥AC，

∴∠AEB=∠CEB，

In △ABE and △CBE.

∠∠ AEB =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠,

∴△ABE≌△CBE，

∴EC=EA，

In Rt△CGE, the square of BG-the square of GE = the square of EA is obtained by Pythagorean theorem.