（2005•嘉兴）顶角为36°的等腰三角形称为黄金三角形．如图，△ABC、△BDC、△DEC都是黄金三角形，已知AB=1，则DE=3−523−52．

Question

（2005•嘉兴）顶角为36°的等腰三角形称为黄金三角形．如图，△ABC、△BDC、△DEC都是黄金三角形，已知AB=1，则DE=

(3−√(5))/(2)

(3−√(5))/(2)

．

(3−√(5))/(2)

(3−√(5))/(2)(3−√(5))/(2)3−√(5)3−√(5)√(5)

5

522

(3−√(5))/(2)

(3−√(5))/(2)(3−√(5))/(2)3−√(5)3−√(5)√(5)

5

522

Answer 1

∵△ABC、△BDC、△DEC都是黄金三角形，AB=1
∴AB=AC，AD=BD=BC，DE=BE=CD，DE∥AB
∴设DE=x，则CD=BE=x，AD=BC=1-x，(DE)/(AB)＝(EC)/(BC)
∴EC=BC-BE=1-x-x=1-2x
∴(x)/(1)＝(1−2x)/(1−x)
解得：DE=(3−√(5))/(2)． (DE)/(AB)DEDEDEABABAB＝(EC)/(BC)ECECECBCBCBC
∴EC=BC-BE=1-x-x=1-2x
∴(x)/(1)＝(1−2x)/(1−x)
解得：DE=(3−√(5))/(2)． (x)/(1)xxx111＝(1−2x)/(1−x)1−2x1−2x1−2x1−x1−x1−x
解得：DE=(3−√(5))/(2)． (3−√(5))/(2)3−√(5)3−√(5)3−√(5)

5

55222．