Ducem OE||AB||CD, E∈BC
In ΔBCD avem :
[tex]\frac{OE}{CD} = \frac{BE}{BC} [/tex]
[tex]\frac{BE}{BC}= \frac{BC-CE}{BC} = \frac{BC}{BC}- \frac{CE}{BC} = 1-\frac{CE}{BC} [/tex]
=>[tex] \frac{OE}{CD} =1- \frac{CE}{BC} [/tex]
dar, In ΔCAB avem:
[tex] \frac{CE}{BC} = \frac{OE}{AB} =\frac{OE}{18} [/tex]
=> [tex] \frac{OE}{CD}=\frac{OE}{6} =1- \frac{OE}{18} [/tex]
=> [tex] \frac{OE}{6} + \frac{OE}{18} =1=\ \textgreater \ \frac{3OE}{18} + \frac{OE}{18} =1=\ \textgreater \ \frac{4OE}{18} =1[/tex]
=> OE=4,5
[tex] \frac{CO}{CA} = \frac{CO}{12} = \frac{OE}{AB} = \frac{OE}{18} = \frac{4,5}{18} [/tex] =>
CO= 3
=> AO=CA-CO= 12-3=9
=> AO=9
