As shown in the figure, it is a diamond trademark designed by Xiaohong. Triangle ABC is an equilateral triangle with a side length of 2, quadrilateral ABDE is an isosceles trapezoid, AC is parallel to ED, and angle EAC. ) It is proved that ∵△ABC is an equilateral triangle,

∴AB=BC, ∠ BAC = ∠ BCA = 6. (1 point)

∵ Quadrilateral ACDE is an isosceles trapezoid, ∠ EAC = 6.

that is, ∠BAE=∠BCD. (2 points)

In △ABE and △BCD, AB=BC, ∠ BAE = ∠ BCD, AE=CD,

∴△ Abe ≌. (6 points)

(3) From (2),

ANCN=

ABCD=2,

∴CN=

12AN=

13AC, (8 points)

similarly AM=

13AC.

∴∠ DCF = 6. (1o points)

In Rt△CDF, ∴∠ CDF = 3,

∴CF=

12CD=

12, < p. (11 points)

In Rt△BDF, ∵BF=BC+CF=2+

12=

52, DF=

32, < P > ∴ BD = < P > BF2+DF2 = < P > (.