It is not true.
Take $A=0$, $B=1$, $C=e^{\frac\pi6 i}$ $D=e^{\frac\pi3 i}$. Then $d=1$ and the perimeter is $2+\tfrac1{\cos\frac{\pi}{12}} >3$.
I am sure that $\frac1{2+\tfrac1{\cos\frac{\pi}{12}}}$ is the optimal bound.
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It is not true. Take $A=0$, $B=1$, $C=e^{\frac\pi6 i}$ $D=e^{\frac\pi3 i}$. Then $d=1$ and the perimeter is $2+\tfrac1{\cos\frac{\pi}{12}} >3$. I am sure that $\frac1{2+\tfrac1{\cos\frac{\pi}{12}}}$ is the optimal bound. |
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It is not true. Take $A=0$, $B=1$, $C=e^{\frac\pi6 i}$ $D=e^{\frac\pi3 i}$. Then $d=1$ and the perimeter is $>3$.2+\tfrac1{\cos\frac{\pi}{12}} >3$. I am sure that $\frac1{2+\tfrac1{\cos\frac{\pi}{12}}}$ is the optimal bound. |
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