[tex]\displaystyle \frac{\overbrace{m(AB)}}{2} = \frac{\overbrace{m(BC)}}{3} \\ \\
\overbrace{m(AC)}= \frac{1}{3} \overbrace{m(BC)}~\Rightarrow~ \frac{\overbrace{m(AC)}}{1} = \frac{\overbrace{m(BC)}}{3} \\ \\
\Rightarrow~\frac{\overbrace{m(AB)}}{2} = \frac{\overbrace{m(BC)}}{3}=\frac{\overbrace{m(AC)}}{1}=k \\ \\ \overbrace{m(AB)}=2k \\ \overbrace{m(BC)}=3k \\ \overbrace{m(AC)}=k [/tex]
[tex]\displaystyle \overbrace{m(AB)}+ \overbrace{m(BC)}+ \overbrace{m(AC)}=360^o \\
2k+3k+k = 360^o \\ 6k=360^o \\
k= \frac{360}{6} = 60 \\
\overbrace{m(AB)} =2k = 2\times 60 = 120^o\\ \overbrace{m(BC)}= 3k=3\times 60 = 180^o\\ \overbrace{m(AC)}=k=60^o \\ \text{Masura unui arc de cerc = unghiul la centru care subantinde arcul.} \\ \text{Unghiurile triunghiului inscris in cerc sunt unghiuri inscrise in cerc. }[/tex]
[tex] \text{Un unghi inscris incerc este egal cu jumatate din unghiul la centru } \\
\text{care subintinde acelasi arc.} \\
Rezulta: \\
\widehat{ACB}= \frac{\overbrace{AB}}{2}= \frac{120}{2}=\boxed{60^o} \\
\widehat{BAC}= \frac{\overbrace{BC}}{2}= \frac{180}{2}=\boxed{90^o} \\
\widehat{ABC}= \frac{\overbrace{AC}}{2}= \frac{60}{2}=\boxed{30^o} \\ [/tex]
