?$證明： (1)連結(jié)OD$?
?$因?yàn)镃D是圓O的切線$?
?$所以CD⊥OD$?
?$所以∠CDO=∠ADC +∠ADO = 90°$?
?$因?yàn)镺A=OD，$?
?$所以∠ADO =∠DAO$?
?$因?yàn)镺F⊥AD于點(diǎn)E$?
?$所以∠DAO+∠AOF = 90°$?
?$所以∠ADC=∠AOF$?
?$(2)設(shè)圓O的半徑為r$?
?$則OA=OB=OD=r$?
?$因?yàn)镃D⊥OD$?
?$所以sinC=\frac {OD}{OC}=\frac {1}{3}$?
?$所以O(shè)C= 3OD= 3r$?
?$所以BC=OC+OB=3r+r=4r$?
?$因?yàn)锳B是圓O的直徑， $?
?$所以∠ADB=90°$?
?$又因?yàn)镺F⊥AD于點(diǎn)E$?
?$所以O(shè)E//BD ，OA= OB$?
?$因?yàn)镺E是△ABD的中位線$?
?$所以O(shè)E=\frac {1}{2}BD$?
?$因?yàn)锽D= 8 $?
?$所以O(shè)E=4$?
?$又因?yàn)镺F//BD$?
?$所以△BCD∽△OCF$?
?$所以\frac {OF}{BD}=\frac {OC}{BC}$?
?$即\frac {OF}{8}=\frac {3x}{4x}$?
?$所以O(shè)F=6$?
?$所以EF= OF- OE=6-4=2$?
