If a and b are real numbers between 0 and 1 such that the points ${\mathrm{z}}_1 = \mathrm{a}+\mathrm{i}, {\mathrm{z}}_2 = 1 + \mathrm{bi}$ and $z_3 = 0$ for, an equilateral triangle then a, b are

$$\begin{gathered} \mathrm{ }0<\mathrm{a}<1,\mathrm{ } 0<\mathrm{b}<1 \\ \left.{\mathrm{z}}_1=(\mathrm{a},1)\mathrm{ } {\mathrm{z}}_2=\mathrm{B}(1,\mathrm{b})\mathrm{ } {\mathrm{z}}_3 = \mathrm{O}(0,0\right) \end{gathered}$$

OAB is an equilateral triangle

$$\begin{gathered} \Rightarrow \mathrm{ }\mathrm{OA}=\mathrm{OB}=A\mathrm{B} \\ \Rightarrow {\mathrm{a}}^2+1= {\mathrm{b}}^2+1\Rightarrow \boxed{\mathrm{a}=\mathrm{b}} \\ and {\mathrm{OA}}^2 = {\mathrm{AB}}^2 \\ \therefore {\mathrm{a}}^2 + 1 = {(\mathrm{a}-1)}^2 + {(1-\mathrm{b})}^2 \\ \Rightarrow {\mathrm{a}}^2 + 1 = 2({\mathrm{a}}^2 - 2\mathrm{a} +1) \\ ={\mathrm{a}}^2-4\mathrm{a}+1=0 \\ \Rightarrow \mathrm{a}=2-\sqrt{3} \mathrm{Since} 0 <\hspace{0.17em}\mathrm{a} <\hspace{0.17em}1 \\ \therefore \mathrm{a} = 2 - \sqrt{3}, \mathrm{b} = 2 - \sqrt{3} \end{gathered}$$