Hypothesis: AC = CB
DE is the diameter
CH is perpendicular to AB (the altitude)
Thesis: DE : BC = BC : CH (BC^2 = DE * CH)
$\displaystyle A=\frac{1}{2} *AB*CH$
$\displaystyle R=\frac{AB*BC*CA}{4A}$ (Circumradius - AoPSWiki)
$\displaystyle R=\frac{AB*BC*CA}{4*\frac{1}{2}*AB*CH}$
$\displaystyle R=\frac{BC*CA}{2CH}$
$\displaystyle R=\frac{BC^2}{2CH}$
$\displaystyle \frac{BC^2}{CH}=2R=DE$
$\displaystyle BC^2=DE*CH$