Pure geometry proof of parabolic mirror property Is there a proof of the property that parallel rays of light incident upon a parabolic mirror converge to its focus that does not resort to Cartesian coordinates?

 A: Yes, there's plenty such proofs, but the choice between them depends on which of the multiple equivalent definitions of a parabola you take, and exactly what restrictions you place on the allowed proofs. 
Physicists normally think of parabolas as the locus of equations of the form $y=x^2$, but you're explicitly looking for something with no formulas, so you probably need to change that as well. The next likeliest candidate is defining the parabola as the locus of points equidistant from a point and a line, which ties you in to all the proofs of classical geometry. Under that understanding, the proof of the reflection property is a staple of euclidean compass-and-straightedge geometry, and Wikipedia has a suitable proof, based on a construction of the form

Image source
In short, with $C$ on the directrix, you define $B$ as the midpoint of $\overline{FC}$, which means (since $\overline{FE}=\overline{EC}$ by definition of the parabola) that $\angle FEB=\angle BEC$, so you just need to show that the line joining $B$ and $E$ is tangent to the parabola. The Wikipedia proof relies on some facts from calculus, though if you want a calculus-free proof you can probably find one by rooting around in the toolbox for the euclidean geometry of conic sections.
