∵ABCD is a square.

∴ab=bc,∠abc=∠dcb=∠ebc=∠bcg=∠fgc=90

∴BCGF is a rectangle.

∴BC=FG,BF=CG,∠BFG=90

∴AB=FG

∵EF⊥AP, then ∠ BAP+∠ AFP = 90.

∠AFP+∠GFE=∠BFG=90

∴∠BAP=∠GFE

∴RT△BAP≌RT△GFE(ASA)

∴BP=EG=CE+CG=CE+BF

2、CE=BF+BP

3、∫BF(AB)∑CE(CD)

∴bp/cp=pf/pe= 1/3(BP∶CP = 1∶3)

Then pf =1/3pe =1/3× 3 √ 5 = √ 5.

∴RT△BFP: BP？ =BF？ -BF？ =(√5)? - 1? =4

BP=2, then CP=3BP=6,

BG=BP=2，FG=BG-BF=2- 1= 1

De: BC=AB=CP-BP=6-2=4。

EF=PE-PF=3√5-√5=2√5

∴RT△PCE: CE=√(PE？ -CP？ )=√[(3√5)? -6? ]=3

RT△ABP: AP=√(AB？ +BP？ )=√(4? +2? )=2√5

∫FG∨CE

∴△FOG∽△COE

∴OF/OE=FG/CE= 1/3

OF/(EF-OF)= 1/3

3OF=(2√5-OF)

OF=√5/2

∫MN∨EF, that is, OF/BN=FG/BG= 1/2.

BN=2OF=√5

∫MN∨PE(EF)

∴∠AMB=∠APE=∠ABP=90

∠∠PAB =∠BAM

∴△ABM∽△APB

∴AB/AP=BM/BP

BM = ab×BP/AP = 4×2/2ì5 = 4ì5/5

∴MN=BM+BN=4√5/5+√5=9√5/5