过正方形ABCD的顶点B作BE‖CA,且使AE=AC,又CF‖AE,求证：∠BCF=1/2∠AEB.

Question

过正方形ABCD的顶点B作BE‖CA,且使AE=AC,又CF‖AE,求证：∠BCF=1/2∠AEB.

Answer 1

过A作AG⊥BE于G,
设AC,BD交于O,则AGBO是正方形,
AG=AO=AC/2=AE/2,又AG⊥GE,所以,∠AEG=30°.
∠CFB=∠AEG=30°,∠FBC=∠FBA+∠ABC=135°,
∠BCF=180°-∠CFB-∠FBC=15°,
∠BCF=1/2∠AEB.