I am reading Maxwell's Matter and Motion and he has this construction as a step in deriving Newton's law of Gravitation from Kepler's First Law.
In this construction $SU$ is equal to the ellipse's major axis $AB$, and $PZ$ is the perpendicular bisector to $HU$. Maxwell states that $HZ \times SY=b^2$, with $b$ being the length of the ellipse's semiminor axis. I can see how this is valid when $HZ=SY$ and I have an idea of how to derive it analytically, but I would like to know how to derive it from properties of the circle and the ellipse, etc. using classical geometry.
Extend $HP$ to meet $SY$ at point $Q$.
$HUQS$ is an isosceles trapezoid and hence a cyclic quadrialteral. Applying Ptolemy's Theorem gives $HZ\cdot SY=\frac {1}{4}\left(AB^{2}-SH^{2}\right)$, which is the length of the semiminor axis.