如图,在Rt△ABC中,∠A=90°,AB=AC=8根号6,点E为AC的中点,点F在底边BC上,且FE垂直BE,则△CEF的面积是多少?

Question

如图,在Rt△ABC中,∠A=90°,AB=AC=8根号6,点E为AC的中点,点F在底边BC上,且FE垂直BE,则△CEF的面积是多少?

Answer 1

∵AE=AC/2=8√6/2=4√6
又∵BE^2=AB^2+AE^2
∴BE=4√30
∵tan∠ABE=AE/AB=1/2
∵tan∠ABC=AC/AB=1
∵tan∠EBF=tan(∠ABC-∠ABE)
=(tan∠ABC-tan∠ABE)/(1+tan∠ABC*tan∠ABE)
=1/3
∴EF=BE*tan∠EBF=4√30/3
∴S△ABC=AB*AC/2=192
∵S△ABE=AB*AE/2=86
∴S△BEF=BE*EF/2=80
∴S△CEF=S△ABC-S△ABE-S△BEF=6