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**