Because ∠ c = 90 and ∠B is a common angle, the triangle DEB is similar to the triangle ACB.

Then BD/AB=DE/AC.

Because in the right triangle ABC, AC=3, BC=4, AB=5.

DE=(4/5)*3= 12/5

2. Proof: connect EO, because OE=OD, so ∠ ODE = ∠ DEO = ∠ A.

Because ∠ EDB+∠ DBE = 90; ∠OED+∠DEF = 90；

So ∠FED=∠DBE

Because ∠ fed+∠ FEA = 90; ∠A+∠DBE = 90；

So ∠ FEA = ∠ A.

So AF=FE

So the triangle FAE is an isosceles triangle.