Geometry puzzle (28)

Problem


Given is quadrilateral ABCD :



Calculate angles x and y.

Solution

These type of problems can be difficult.
I do not know about a general approach.
Some intuition is required beside trial and error and perseverance.

First we construct the circumscribed circle of triangle BCE.
DC and AB are extended, their intersection is G.
H is the intersection of DG and the circle.



ΔABD....LABD = 64.
ΔACD....LACD = 74.
ΔABE....LAEB = 68

x + y = 68....{exterior angle of ΔBCE}
arcBC = 2x+2y = 136
ΔAGD....LG = 24.
(arcBC - arcFH)/2 = 24
arcFH = 88.

Construct the perpendicular bisector of BC.
This line intersects the circle in H' and the extended line AG in P.



LP = 34 - 24 = 10.
10 = (arcFH'- arcBK)/2.
20 = arcFH' - 68.
arcFH = arcFH' = 88.

Points H and H' coincide.

y = 180 - LACD - 56.
y = 50.
x = 18.