Question

如图,BC为半圆O的直径,AD垂直于BC,垂足为D,过点B做弦BF交AD于点E,交半圆O于点F,弦AC与BF交于点H且A

Answer 1

证明：（1）∵AE=BE,

∴∠BAD=∠ABE,

∵BC是直径,AD⊥BC,

∴∠ADB=∠BAC=90°,

∴∠ABD+∠BAD=∠ABC+∠C=90°,

∴∠BAD=∠C,

∴∠C=∠ABF,

∴AB＝AF

；

（2）∵∠C=∠ABF,

Rt△ABH∽Rt△ACB,

∴AH：BH=AB：BC,即AH•BC=AB•BH,

∵∠EAH+∠BAD=∠AHB+∠ABH=90°,∠BAD=∠ABE,

∴∠EAH=∠AHB,

∴AE=EH=BE=1 /2 BH,