Geometry Help Urgently?

Well...I don't know how "fast" this is

Let   B = (0, 0)   A = (0, 2)    E = (0, 1)   F = (1,0)  D = (2, 2)

We can find the equation of the line containing AF as   y = -2x + 2

And the equation  of the line containing  BD is  y =  1x

So we can find the x coordinate of H as

x = -2x + 2

3x = 2

x = 2/3

And y = 2/3

So...the area of triangle BHF =  (1/2) (BF)(2/3)  = (1/2) (1) (2/3) = 1/3

And the equation of the line containing ED is (1/2)x + 1

So....the x coordinate of I can be found as  

(1/2)x + 1 = -2x+ 2

(5/2)x = 1

x = 2/5

And the y coordinate = (1/2)(2/5) + 1 =  1/5 + 1 = 6/5

Now....the area of triangle ADE = (1/2)(AE)( AD)  = (1/2)((1) (2) =   1

And the area of triangle ADI = (1/2)(height) (AD) =  (1/2) (2 - 6/5) (2)  =  (1/2)(4/5)(2)  = 4/5

So...the area  of triangle AEI = area of  ADE - area of ADI =   1 - 4/5 = 1/5

And the area of triangle ABF = (1/2)(BF)(AB) = (1/2)( 1) (2) =  1

So...the area of BEIH =  area of ABF - area of AEI - area of BHF =

1 - (1/5) - (1/3)  =

1 - 3/15 - 5/15 =

1 - 8/15 =

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