Here is a quick method to solve the puzzle by mirroring and looking for similar triangles.

1. - From point B we draw a line BD1 which a mirror of CD with respect of OA bisector of angle â. So BD1 will form obviously 60 degrees angle with respect to base BC like is the case for CD

2. - BD1 intersect to CD in P, so being 60 degrees two of the angles of triangle BPC the angle in P must be 60 degrees too, so BPC is equilateral and is side values r the radius of the circumference. So Cp equals r.

3. - Since the arc EP of the circumference is common to the angles ECP and angle EBP and since point C lays at the circumference and B at its center, the central angle EBP amounts twice the value of angle ECP what is 40 degree so the triangle BEF is equilateral too and its side EF equals r also. So point M of intersection of EF and CD belongs to the bisector line of angle CBF.

4. - By mirroring with respect to line BM the triangle MDF we get MD2C. It's immediate to find, since angle ED2C equals 40 degrees, that D2EC equal also 40 degree which means that D2CE is isosceles and side CE equals to CD2 which is the mirror of FD. (q.e.d)

1 Point D2, as it can be seen in the drawing is at the extension of the base line BC, just to the left of C.

2 As is stated “By mirroring with respect to line BM the triangle MDF we get MD2C” , as a consequence angle CD2M is equal to angle MDF which values 40 degree.

Now by projecting E on CB the base of the original triangle let us call E1 this projection, it is easy to follow that the angle D2EE1 is complementary of CD2E what means has a value 50 degrees.
Now if we subtract to the angle D2EE1 the angle E1EC (equal to 10 degrees since is equal to the bisector of BAC) we get that D2EC equal to 40 degrees also what forces the triangle D2CE to be isosceles.

I do apologize for stating erroneously in the drawing CEE1 equals to 20 degrees when it'should read 10 degrees, as I said before in transferring process from my hand sketch I committed some errors …. I am struggling with this new graphic cad software. If you cannot get my explanation I am ready to make another more clarifying drawing.

Sorry for my logic. English is not my mother language besides I am not a mathematician but just a kind of old fashion engineer.

Thank you.