= Stationary curves of the planar isoperimetric problem
{title2=$(x-c_2)^2+(y+c_1)^2=\lambda^2$}
For a regular closed planar curve, maximize signed enclosed area at fixed length using the variational integrand $\mathcal L=(x\dot y-y\dot x)/2+\lambda\sqrt{\dot x^2+\dot y^2}$. Write the nonzero speed as $v$. The <Euler-Lagrange equations> integrate to $\lambda\dot x/v=y+c_1$ and $\lambda\dot y/v=-x+c_2$. A zero multiplier would force a zero velocity, so squaring and adding yields the displayed circle. The argument uses only regularity and is independent of the speed used to trace the curve. A positively oriented simple traversal of prescribed length $L$ encloses area $L^2/(4\pi)$; the <planar isoperimetric inequality> supplies the separate global maximality statement.
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