= Conformal modulus of a quadrilateral
{title2=$M=w/h$}
Map a <conformal quadrilateral> to a rectangle of width $w$ and height $h$, with the chosen opposite boundary arcs mapped to its vertical sides. Their <extremal length> is $M=w/h$; the other pair has <extremal length> $1/M$. The constant <conformal metric> is extremal. For an arbitrary density, average horizontal joining lengths over height and apply the <Cauchy-Schwarz inequality> to obtain $L_\rho^2h\leq wA_\rho$, proving the upper bound. Vertical joining segments prove the dual formula. <Conformal invariance of extremal length> transports both equalities back to the quadrilateral.
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