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The most difficult problem in mathematics
As shown in figure (1), in quadrilateral ABCD, AB = BC = CD, ∠ ABC = 78, ∠ BCD = 162. Let the extension lines of AD and BC intersect at point E, then ∠ AEB = _ _ _. (Shanghai in 2009

Answer: 2 1

Solution (not in the original counseling materials, here is the solution process I did):

As shown in Figure (2), BC and AB respectively intersect at point F through parallel lines of point A and point C to connect DF.

∫∠ABC = 78,ab∨cf ┃∴cd=cf=af

∴∠ BCF = 102∴∴△ CDF is a regular triangle.

∫≈BCD = 162 ┃∴af=df,∠cfd=∠cdf=60

∴∠DCF=60,∠DCE = 18 ┃∴∠adf=﹙ 180 -∠afd﹚÷2=2 1

∫ab∨cf,BC∨af,AB=BC ┃∴∠ADC=39

∴ quadrilateral ABCF is a diamond, ∠ AFC = 78┃∵∠ ADC is the outer corner of △DCE, ∠ DCE = 18.

∴BC=CF=AF ┃ ∴∠AEB=2 1

bc = cd

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This is a competition problem in 2009. I hope the O(∩_∩)O~ dug up in my tutoring materials will help.