  A streamlined proof that the Bergman kernel associated to a quadrature domain
in the plane must be algebraic will be given. A byproduct of the proof will be
that the Bergman kernel is a rational function of z and one other explicit
function known as the Schwarz function. Simplified proofs of several other well
known facts about quadrature domains will fall out along the way. Finally,
Bergman representative coordinates will be defined that make subtle alterations
to a domain to convert it to a quadrature domain. In such coordinates,
biholomorphic mappings become algebraic.
