= Symmetry rule for extremal length
{title2=$\lambda(\Gamma)=\sup_{T^*\rho=\rho}L_\rho(\Gamma)^2/A_\rho$}
If a <conformal map> or <anticonformal map> $T$ is an <involution> preserving a curve family, its <extremal length> can be computed using invariant <conformal metrics>. Replace any density by $\rho_s=(\rho+T^*\rho)/2$. Every curve and its image have length at least $L_\rho$, hence $L_{\rho_s}\geq L_\rho$. Convexity of the square and <change of variables> give $A_{\rho_s}\leq A_\rho$, proving the assertion. For reflection in the real axis, invariant means $\rho(z)=\rho(\bar z)$.<Conformal Maps and Geometry by Dmitry Beliaev (2015 lecture notes)>[Proposition 4.3.8, printed pages 52–53: restriction of the supremum to symmetric metrics]{ref}
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