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Ib Math IA 
These are the extra construction lines required to solve this problem. Here is the step by step thought process:
Find the hypotenuse of triangle AGC.
Use the circle theorem angles in the same segment are equal to show that angle CBD = angle CAG.
Therefore triangle AGC and GBD are similar, so length BG = 4. We can now use Pythagoras to find length BD.
We can find length CD by Pythagoras.
Now we have 3 sides of a triangle, CDB. This allows use to find angle BDC using the cosine rule.
Now we the circle theorem angles in the same segment are equal to show that angle BDC = angle BEC.
Now we use the circle theorem angles in a semi circle are 90 degrees to show ECB = 90.
Now we have a right angled triangle BCE where we know both an angle and a side, so can use trigonometry to find the length of BE.
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