Find the legnths x and y     Thanks for your help

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This problem can be solved using laws of similar triangles.  The ratios of corresponding sides of similar triangles are equal.  This also includes corresponding segments in similar triangles.

To find x:

Step 1:  Write equal ratios as a proportion.

AG / AF = BC / BD
14 / (14 + 21) = x / (x + 15)
14 / 35 = x / (x + 15)

Step 2:  Solve the proportion by cross-multiplying.

14 * (x + 15) = 35 * x
14x + 210 = 35x
14x + 210 + (-14x) = 35x + (-14x)
210 = 21x
210 / 21 = 21x / 21
x  = 10

To find y:

Step 1:  Write equal ratios as a proportion.

FE / AE  = DE / BE 
y / (14 + 21 + y) = 20 / (10 + 15 + 20)
y / (35 + y) = 20 / 45

Step 2:  Solve the proportion by cross-multiplying.

45 * y = 20 * (35 + y) 
45y = 700 + 20y
45y + (-20y) = 700 + 20y + (-20y)
25y = 700
25y / 25 = 700 / 25
y = 28

Final answer:
x = 10
y = 28

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