Mgccl's Ivory Towerso tired

The World's Hardest Easy Geometry Problem ruined my sleep.

Find the measure of without elementary geometry about triangles.

I tried that problem a few months ago for a few hours and failed, so I cheated and looked up the answer.
But today I dreamed about that problem. In my dream, I saw myself using circle to solve that problem.
I woke up, it's 3 am, great time to work on a geometry problem.

So I did. I succeed!
It is cheating to use circles. Only basic properties about triangles, angles and lines are allowed.
Please don't read this if you want to have your own fun with the problem first.
I also have a link for a triangle only solution.

Note: My graph is different because I changed the name of point E to D, and D to E.
Here is where da solutions at.

First looking show that , so it's nice to consider using symmetry to solve the problem.
Construct
Draw a circle pass through
Reflect all points through
The graph looks like this:

It's easy to see there are some equations

All the angles listed above can be found directly because the other two angles in the triangle are already known.
After finding all the size of those angles and multiply 2 to each side, it reduce to a few simple linear equation, all we have to do is to solve it.
It's a easy linear system can be computed by hand.
Set

We end up with this

because the graph is symmetric.

Thus

Q.E.D.[Quite Exhausting Demonstration]

I need to start to train my subconscious so I can do math when I'm asleep.