數學~~證明題目~How to prove?

1) AC and BD are diagonals of a square, thus ∠AOF = ∠BOE = 90 deg.

Also AO = BO (Diagonals of square bisects each other)

Then with ∠BGF = ∠BOE = 90 deg. and ∠GBF = ∠OBE (Vert. opp. ∠s), △GBF and △OBE are similar and hence ∠OEB = ∠OFA

Hence △BOE and △AOF are congurent (AAS)

2) Let AD = AB = x and BE = CF = DG = y

With BEPF being a rectangle, PF = BE = y and hence:

EF2 = (x - y)2 + y2

With CG = x - y, we have:

FG2 = (x - y)2 + y2

Thus EF = FG

Then with △GFC and △FEB are congurent (SAS): ∠GFC = ∠FEB and ∠FGC = ∠EFB

Also ∠GFC + ∠FGC = 180 - ∠GCF = 90 deg., so ∠GFC + ∠EFB = 90 deg., making ∠EFG = 90 deg.

Finally EF and FG are perp.