Because ∠ ABC = 45, and CA⊥BF and △ABC are isosceles straight triangles, AB=AC①.

Because ∠ F+∠ FBD = 90, ∠ F+∠ FCA = 90.

So ∠FBD=∠FCA②

Because △ABD and △ACF are right triangles, ① and ② are combined.

So Rt△ABD is equal to Rt△ACF.

So BD=FC

Because the BE bisector ∠ABC is perpendicular to AC, the triangle ABC is an isosceles triangle, so e is the midpoint of AC.

So CE= 1/2FC= 1/2BD.

That is BD=2CE.