Question

If $a$ and $b$ are real numbers between $0$ and $1$ such that the points $z_1=a+i,z_2=1+bi$ and $z_3=0$ from an equilateral triangle, then find the values of ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$.

Answer 1

${|z_1-z_2|}^2={(a-1)}^2+{(1-b)}^2{|z_2-z_3|}^2=1+b^2{|z_3-z_1|}^2=a^2+11+b^2=a^2+1⟹a^2=b^2⟹a=b{a}^2+1={(a-1)}^2+{(1-b)}^2=2{(a-1)}^2=2a^2-4a+2⟹0=a^2-4a+1⟹a=2\pm \sqrt{3}\therefore a=b=2-\sqrt{3}$