geometry puzzle

Geometry puzzle (14)

Asked: find angles x, y at point D in above picture.

Solution

Draw a circle with center E and radius EB.
Extend some lines, see picture below.

It seems that point D is part of the circle.
That's what we have to prove.

Please look at ΔABE.
LLEB = 40 because of being an exterior angle.
Arc BI = 40 = arc FG.
LCBE = 20....arc FD = 40.
LDEI = 100 = arc DI.
LDAC = 80 = (arc DI + arc GH)/2....arc GH = 60.
arc HB = 360 - (40 + 100 + 40 + 20 + 20 + 60) = 80.
Look at ΔABD.
x + y = 180 - (110 + 30) = 40.
Point D is part of the circle because LD is 40 degrees and spans arc HB = 80.
Because DE = EB, LBDE = 20 = y = x.