$\beta ={\displaystyle \frac{\text{m}\stackrel{⌢}{\text{AC}}-\text{m}\stackrel{⌢}{\text{DF}}}{2}}$

$\text{m}\mathrm{\angle }\text{ABC}={\displaystyle \frac{\text{m}\mathrm{\angle }\text{AOC}-\text{m}\mathrm{\angle }\text{FOD}}{2}}$

$\alpha ={\displaystyle \frac{\text{m}\stackrel{⌢}{\text{AC}}+\text{m}\stackrel{⌢}{\text{DF}}}{2}}$

$\text{m}\mathrm{\angle }\text{DEF}={\displaystyle \frac{\text{m}\mathrm{\angle }\text{AOC}+\text{m}\mathrm{\angle }\text{FOD}}{2}}$

Segments From Secants

If two secants are drawn from a common point outside a circle then $a(a+b)=c(c+d)$